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Trade-off between recognition an reconstruction: Application of Robotics Vision to Face Recognition
 

Trade-off between recognition an reconstruction: Application of Robotics Vision to Face Recognition

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Autonomous and ecient action of robots requires a robust robot vision system that can ...

Autonomous and ecient action of robots requires a robust robot vision system that can
cope with variable light and view conditions. These include partial occlusion, blur, and
mainly a large scale di erence of object size due to variable distance to the objects. This
change in scale leads to reduced resolution for objects seen from a distance. One of the
most important tasks for the robot's visual system is object recognition. This task is also
a ected by orientation and background changes. These real-world conditions require a
development of speci c object recognition methods.
This work is devoted to robotic object recognition. We develop recognition methods
based on training that includes incorporation of prior knowledge about the problem.
The prior knowledge is incorporated via learning constraints during training (parameter
estimation). A signi cant part of the work is devoted to the study of reconstruction
constraints. In general, there is a tradeo between the prior-knowledge constraints and
the constraints emerging from the classi cation or regression task at hand. In order to
avoid the additional estimation of the optimal tradeo between these two constraints, we
consider this tradeo as a hyper parameter (under Bayesian framework) and integrate
over a certain (discrete) distribution. We also study various constraints resulting from
information theory considerations.
Experimental results on two face data-sets are presented. Signi cant improvement in
face recognition is achieved for various image degradations such as, various forms of image
blur, partial occlusion, and noise. Additional improvement in recognition performance is
achieved when preprocessing the degraded images via state of the art image restoration
techniques.

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    Trade-off between recognition an reconstruction: Application of Robotics Vision to Face Recognition Trade-off between recognition an reconstruction: Application of Robotics Vision to Face Recognition Document Transcript

    • TEL-AVIV UNIVERSITY The Iby and Aladar Fleischman Faculty of EngineeringTRADE-OFF BETWEEN RECOGNITION AND RECONSTRUCTION: APPLICATION OF NEURAL NETWORKS TO ROBOTIC VISION Thesis submitted for the degree ”Doctor of Philosophy” by INNA STAINVAS Submitted to the Senate of Tel-Aviv University 1999
    • TEL-AVIV UNIVERSITYThis work was carried out under the supervision of Doctor Nathan Intrator and Doctor Amiram Moshaiov
    • This work is dedicated to my family
    • AcknowledgmentI would like to thank my husband, daughter and parents for their tolerance and moralsupport during the completion of this thesis. I am greatly indebted to my first advisor Dr. Amiram Moshaiov, who gave me achance to start as a Ph.D. Student at the Engineering Faculty of Tel-Aviv University,when I was only two months in Israel. I am very grateful to him for proposing to work inNeural Networks and Computer Vision and for allowing me freedom in my research. I have been pleasantly surprised by the flexibility of the educational system of the Tel-Aviv University in allowing me to listen and participate in courses at different faculties,such as the Engineering Faculty, Computer Science and Foreign Languages. While taking courses in Neural Networks, I met Dr. Nathan Intrator, who became mymain supervisor and collaborator for more than five years. He opened me to a new worldof Neural Networks and I have learned much from him, not only on the technical aspectsbut also on scientific research methodologies. Without him, this thesis would have neverappear. I am grateful to him for his tolerance, endless support and guidance. It is impossible to thank all the people who helped me, but I would like to mention thesystem administrator of the Engineering faculty, Udi Mottelo, the Department secretaryAriella Regev, the secretary of the Emigration Support department Ahuva, my friends,and the people of the Neural Computation Group of Computer Science faculty, YairShimshoni, Nurit Vatnick and Natalie Japkowich. This work was supported by grants from the Rich Foundation, the Don and SaraMarejn Scholarship Fund and by a grant from the Ministry of Science to Dr. NathanIntrator. Inna StainvasMarch 8, 1999
    • AbstractAutonomous and efficient action of robots requires a robust robot vision system that cancope with variable light and view conditions. These include partial occlusion, blur, andmainly a large scale difference of object size due to variable distance to the objects. Thischange in scale leads to reduced resolution for objects seen from a distance. One of themost important tasks for the robot’s visual system is object recognition. This task is alsoaffected by orientation and background changes. These real-world conditions require adevelopment of specific object recognition methods. This work is devoted to robotic object recognition. We develop recognition methodsbased on training that includes incorporation of prior knowledge about the problem.The prior knowledge is incorporated via learning constraints during training (parameterestimation). A significant part of the work is devoted to the study of reconstructionconstraints. In general, there is a tradeoff between the prior-knowledge constraints andthe constraints emerging from the classification or regression task at hand. In order toavoid the additional estimation of the optimal tradeoff between these two constraints, weconsider this tradeoff as a hyper parameter (under Bayesian framework) and integrateover a certain (discrete) distribution. We also study various constraints resulting frominformation theory considerations. Experimental results on two face data-sets are presented. Significant improvement inface recognition is achieved for various image degradations such as, various forms of imageblur, partial occlusion, and noise. Additional improvement in recognition performance isachieved when preprocessing the degraded images via state of the art image restorationtechniques.
    • Contents1 Introduction 1 1.1 General motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 Robotic vision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.2 Internal data representation . . . . . . . . . . . . . . . . . . . . . . 2 1.1.3 Data compression . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1.4 Face recognition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Overview of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 Statistical formulation of the problem 8 2.1 Bias-Variance error decomposition for a single predictor . . . . . . . . . . . 9 2.2 Variance control without imposing a learning bias . . . . . . . . . . . . . . 10 2.3 Variance control by imposing a learning bias . . . . . . . . . . . . . . . . . 12 2.3.1 Smoothness constraints . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.3.2 Invariance bias constraints . . . . . . . . . . . . . . . . . . . . . . . 13 2.3.3 Specific bias constraints . . . . . . . . . . . . . . . . . . . . . . . . 14 2.4 Reconstruction bias constraints . . . . . . . . . . . . . . . . . . . . . . . . 16 2.5 Minimum Description Length (MDL) Principle . . . . . . . . . . . . . . . . 17 2.5.1 Minimum description length . . . . . . . . . . . . . . . . . . . . . . 19 2.6 Bayesian framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.7 MDL in the feed-forward NN . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.7.1 MDL and EPP bias constraints . . . . . . . . . . . . . . . . . . . . 24 2.8 Appendix to Chapter 2: Regularization problem . . . . . . . . . . . . . . . 283 Imposing bias via reconstruction constraints 30 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.1.1 Principal Component Analysis (PCA) . . . . . . . . . . . . . . . . . 30 3.1.2 Autoencoder network and MDL . . . . . . . . . . . . . . . . . . . . 31 3.1.3 Reconstruction and generative models . . . . . . . . . . . . . . . . 34 i
    • 3.1.4 Classification via reconstruction . . . . . . . . . . . . . . . . . . . . 35 3.1.5 Other applications of reconstruction . . . . . . . . . . . . . . . . . . 38 3.2 Imposing reconstruction constraints . . . . . . . . . . . . . . . . . . . . . . 38 3.2.1 Reconstruction as a bias imposing mechanism . . . . . . . . . . . . 38 3.2.2 Hybrid classification/reconstruction network . . . . . . . . . . . . . 40 3.2.3 Hybrid network and MDL . . . . . . . . . . . . . . . . . . . . . . . 40 3.2.4 Hybrid network as a generative probabilistic model . . . . . . . . . 43 3.2.5 Hybrid Neural Network architecture . . . . . . . . . . . . . . . . . . 44 3.2.6 Network learning rule . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.2.7 Hybrid learning rule. . . . . . . . . . . . . . . . . . . . . . . . . . . 484 Imposing bias via unsupervised learning constraints 50 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 4.2 Information principles for sensory processing . . . . . . . . . . . . . . . . . 51 4.3 Mathematical background . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.3.1 Entropy maximization (ME) . . . . . . . . . . . . . . . . . . . . . . 53 4.3.2 Minimization of the output mutual information (MMI) . . . . . . . 55 4.3.3 Relation to Exploratory Projection Pursuit. . . . . . . . . . . . . . 57 4.3.4 BCM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 4.3.5 Sum of entropies of the hidden units . . . . . . . . . . . . . . . . . 59 4.3.6 Nonlinear PCA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4.3.7 Reconstruction issue . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4.4 Imposing unsupervised constraints . . . . . . . . . . . . . . . . . . . . . . . 61 4.5 Imposing unsupervised and reconstruction constraints . . . . . . . . . . . . 625 Real world recognition 69 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 5.1.1 Face recognition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 5.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 5.2.1 Different architecture constraints . . . . . . . . . . . . . . . . . . . 75 5.2.2 Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 5.2.3 Neural Network Ensembles . . . . . . . . . . . . . . . . . . . . . . . 80 5.2.4 Face data-sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 5.2.5 Face normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 5.2.6 Learning parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 82 5.3 Type of image degradations . . . . . . . . . . . . . . . . . . . . . . . . . . 83 5.4 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 ii
    • 5.4.1 Different architecture constraints and regularization ensembles . . . 86 5.5 Saliency detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 5.5.1 Saliency map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 5.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 5.7 Appendix to Chapter 5: Hidden representation exploration . . . . . . . . . 956 Blurred image recognition 100 6.1 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 6.1.1 Experimental design . . . . . . . . . . . . . . . . . . . . . . . . . . 101 6.2 Image degradation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 6.2.1 Main filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 6.2.2 Other types of degradation . . . . . . . . . . . . . . . . . . . . . . . 106 6.3 Image restoration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 6.3.1 MSE minimization and regularization . . . . . . . . . . . . . . . . . 107 6.3.2 Image restoration in the frequency domain . . . . . . . . . . . . . . 109 6.3.3 Denoising . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 6.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 6.4.1 Image filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 6.4.2 Classification of noisy data . . . . . . . . . . . . . . . . . . . . . . . 114 6.4.3 Gaussian blur . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 6.4.4 Motion blur . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 6.4.5 Blind deconvolution . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 6.4.6 All training schemes . . . . . . . . . . . . . . . . . . . . . . . . . . 120 6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1217 Summary and future work 124 7.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 7.2 Directions for future work . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 iii
    • List of Figures 2.1 Supervised feed-forward network . . . . . . . . . . . . . . . . . . . . . . . . 23 2.2 Hybrid network with EPP constraints . . . . . . . . . . . . . . . . . . . . . 25 3.1 Autoencoder network architecture . . . . . . . . . . . . . . . . . . . . . . . 32 3.2 Eigenspaces extracted by PCA . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.3 Combined recognition/reconstruction network . . . . . . . . . . . . . . . . 40 3.4 Hybrid network with reconstruction and EPP constraints . . . . . . . . . . 41 3.5 Detailed architecture of the recognition/reconstruction network . . . . . . 45 4.1 Feed-forward network for independent component extraction . . . . . . . . 53 4.2 Pdf’s graphs for a family of the exponential density functions . . . . . . . . 65 4.3 Exploratory projection pursuit network . . . . . . . . . . . . . . . . . . . . 66 5.1 Misclassification rate time evolution . . . . . . . . . . . . . . . . . . . . . . 77 5.2 MSE (mean-squared) recognition error time evolution . . . . . . . . . . . . 78 5.3 Classification based regularization . . . . . . . . . . . . . . . . . . . . . . . 79 5.4 “Caricature” faces in three resolutions . . . . . . . . . . . . . . . . . . . . 81 5.5 Image degradation and reconstruction (TAU data-set) . . . . . . . . . . . . 84 5.6 Summary of different networks and different image degradations . . . . . . 90 5.7 Saliency map construction . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 5.8 Hidden unit activities vs. classes - for an unconstrained network . . . . . . 96 5.9 Hidden unit activities vs. classes - for a reconstruction network . . . . . . . 97 5.10 Pdf ’s of the hidden unit activities . . . . . . . . . . . . . . . . . . . . . . 98 5.11 Hidden weight representation . . . . . . . . . . . . . . . . . . . . . . . . . 99 6.1 Experimental design schemes . . . . . . . . . . . . . . . . . . . . . . . . . . 102 6.2 Training scheme C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 6.3 Degraded images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 6.4 Noisy Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 6.5 Gaussian blur and restoration . . . . . . . . . . . . . . . . . . . . . . . . . 116 iv
    • 6.6 Motion blur and deblur . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1186.7 Blind deconvolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1196.8 Recognition of blurred images via schemes A–C . . . . . . . . . . . . . . . 1206.9 Reconstruction of Gaussian blurred images . . . . . . . . . . . . . . . . . . 123 v
    • List of Tables 4.1 Unsupervised constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 5.1 Classification results for Pentland data-set . . . . . . . . . . . . . . . . . . 85 5.2 Different ensemble types (Pentland data-set) . . . . . . . . . . . . . . . . . 87 5.3 Different ensemble types (TAU data-set) . . . . . . . . . . . . . . . . . . . 88 5.4 Recognition using saliency map (Pentland data-set) . . . . . . . . . . . . . 92 5.5 Recognition using saliency map (TAU data-set) . . . . . . . . . . . . . . . 93 6.1 Classification results for filtered data . . . . . . . . . . . . . . . . . . . . . 112 6.2 Noise and restoration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 6.3 Gaussian blur and restoration . . . . . . . . . . . . . . . . . . . . . . . . . 115 6.4 Motion blur and restoration . . . . . . . . . . . . . . . . . . . . . . . . . . 117 6.5 Blind deconvolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 6.6 Blurred image recognition via joined ensembles . . . . . . . . . . . . . . . . 121 7.1 Classification error for reconstructed images . . . . . . . . . . . . . . . . . 127 vi
    • Chapter 1Introduction1.1 General motivation1.1.1 Robotic visionNowadays, robots that can move and operate autonomously in a real-world are in highdemand. One of the main perception tasks that has to be addressed in this context is arecognition task. The recognition task in a real-world environment is challenging as it hasto address data variability, such as orientation, changing background, partial occlusionand blur, etc. For illustration let us consider a vision-guided robot helicopter which has to navigateautonomously using only on-board sensors and computing power (Chopper, 1997). Oneof the basic difficulties in recognition of images taken by helicopter cameras during an op-eration is the significant difference between these images and the images, which the robotis acquainted with in ideal flight conditions. Usually, the images taken during operationcontain a large amount of degradation caused by diverse factors, such as illuminationchanging, bad weather conditions, relative motion between the cameras and the objectof interest in the scene, shadows and, low resolution capacity of the cameras, etc. Someof these factors cause images to look blurred and foggy, others lead to noise and partialocclusion. All these factors are crucial for recognition performance and require specialcare. Among the possible approaches to improve recognition performance of degraded im-ages is an endeavor to recover images using state of the art restoration techniques aspreprocessing before a recognition stage. This preprocessing requires estimation of thedegradation process, e.g. the type and parameters of the blur operation. Another ap-proach is to directly address the variability in the recognition system. It is well knownthat for a restoration process to be successful a degradation process has to be accurately 1
    • Chapter 1: Introduction 2modeled. However, in many cases, an exact modeling is impractical, and the restored im-ages remain partially degraded and contain artifacts. Furthermore, restoration methodsare often computationally expensive and require a-priori knowledge or human interaction.It follows that efforts have to be concentrated on development of recognition methods thatare more robust to image degradations.1.1.2 Internal data representationAn important aspect of robust recognition methods is construction of an internal datarepresentation (feature extraction), that captures the significant structure of the data.According to D. Marr (1982) finding an internal representation is an inherent componentof the vision process.Feature based representation Many recognition methods include grouping or per-ceptual organization as a first stage of the visual processing. In this stage, objects arerepresented as models, containing the essential features and logic tight rules needed forrecognition. Some methods extract “anchor points” (Ullman, 1989; Brunelli and Poggio,1992), others consider edge segments as interesting feature elements (Bhanu and Ming,1987; Liu and Srinath, 1984). A relatively new approach is a deformable template match-ing (Grenander, 1978; Brunelli and Poggio, 1993; Jain et al., 1996) and using generalizedsplines for object classification (Lai, 1994). These methods attempt to extract salientfeatures locally in the low level stage of the visual processing, according to subjectiveunderstanding of an investigator. Therefore, finding an internal representation based onextraction of object features and relation between them may be limited.Learning internal representations via Neural Networks A radical alternativeapproach is to use all the available intensity information for finding internal representation.Principal Component Analysis (PCA) (Fukunaga, 1990) is a non neural network exampleof this approach, where internal representation space is spanned by the largest eigenvectorsof the data covariance matrix. These eigenvectors are macro-features extracted implicitlyfrom the images. When fed with intensity images, Neural Networks similar to PCA extractinternal representation in the space of hidden unit activities. Processing an image as a whole is a high dimensional recognition task that leads tothe curse of dimensionality (Bellman, 1961) which means that there is not enough datato robustly train a classifier in a high dimensional space. As an example, a network witha single hidden unit and input images of 60 × 60 pixels has 3600 weight parameters thathave to be estimated. Thus, the main issue is finding an intrinsic low dimensional
    • Chapter 1: Introduction 3representation of the images. As was pointed out by Geman et al. (1992), a way to avoidthe curse of dimensionality in Neural Networks is to prewire the important generalizationsby purposefully introducing learning bias. The work presented in this thesis is specifically devoted to this issue. We developimage recognition techniques using hybrid feed-forward Neural Networks, obtained byintroducing a learning bias. In particular, we investigate the influence of the novel re-construction learning constraints on the recognition performance of feed-forward NeuralNetworks. In addition, we propose to use other learning constraints based on informationtheory, and subsequently compare their efficiency with reconstruction learning constraints.We demonstrate that hybrid Neural Networks are robust to real-world degradation in theinput visual data and show that their performance can be further enhanced when stateof the art (deblur) techniques are also incorporated.1.1.3 Data compressionOften, a compression goal is defined as finding a compact data representation leading togood data reconstruction. Principal Component Analysis (PCA), Discrete Fourier Trans-form (DFT) and its generalization, Wavelet Transform and advanced best basis repre-sentations (Coifman and Wickerhauser, 1992), are examples of compression techniques.Compression may be also realized via an autoencoder network (Cottrell et al., 1987). Theautoencoder is a multi layer perceptron (MLP) type of the network with the output layercoinciding with the input layer and a hidden layer of a small size. Recently a novel type of an autoencoder network has been proposed by Zemel (1993).The hidden layer is allowed to have a large number of hidden units but it has different con-straints on the developed hidden representation. The network is simultaneously trainedto accurately reconstruct the input and to find a succinct representation in the hiddenlayer, assuming sparse or population code formation in the autoencoder hidden layer. When the main task is recognition, the compressed data representation has been usedinstead of the original (high-dimensional) data (Kirby and Sirovich, 1990; Turk and Pent-land, 1991; Murase and Nayar, 1993; Bartlett et al., 1998). Recognition from this rep-resentation is faster and may have better generalization performance. However, it isclear, that such compression is task-independent and may be inappropriate for a specificrecognition task (Huber, 1985; Turk and Pentland, 1993). We seek a compact data description that is task-dependent, and is good for recognition.Thus, the quality of the compression scheme is judged by its generalization property.Often, a separate low-dimensional representation is created for every specific task at hand.Another strategy could be to discover a hidden representation that is suitable for several
    • Chapter 1: Introduction 4potential visual tasks (Intrator and Edelman, 1996). We show that a good task-dependentcompression is obtained when the data representation is constructed not only to minimizethe mean-squared recognition error, but also to maintain data fidelity and/or to extractgood statistical properties. These good properties may be the independence of hiddenneurons, maximum information transfer in the hidden layer or a multi-modal distributionof the hidden unit activities. Therefore, in this case compression is task-dependent and isassisted by the a-priori knowledge. In summary, we investigate lossy compression techniques based on the two visual tasks- image recognition and reconstruction. Our goal is to find a hidden representation thatoptimizes the recognition using hints of the reconstruction task.1.1.4 Face recognitionThe performance of the proposed recognition schemes is examined on two facial datasets. Face recognition has gained much attention in recent years due to the variety ofcommercial applications, such as video conferencing, security, human communication androbotics. Face recognition has recently attracted special attention of different humanrobotic groups, that intensively work on the creation of personal adaptive robots to assistthe frail and elderly blind people, and creation of working mobile robots for deliveryassistance (Hirukawa, 1997; Connolly, 1997). This recognition task is a very difficult one (Chellapa et al., 1995), since it is a high di-mensional classification problem leading to “curse of dimensionality”. This is complicatedby the large variability of the facial data sets due to: • viewpoint dependence • nonrigidity of the faces • variable lighting conditions • motionThe task of face recognition is a particular case of the learning when the variability ofthe data describing the same class is comparable with the similarity between differentclasses. Other important possible recognition tasks from the same category may be therecognition of different kinds of tanks, ships, planes and cars, etc.
    • Chapter 1: Introduction 51.2 Overview of the thesisThe thesis focuses on developing Neural Network techniques that improve the recognitionperformance. A key aspect of this work is finding data representations that lead to bettergeneralization. We show that networks which are trained to recognize and reconstructimages simultaneously extract features that improve recognition. Improved performanceis also achieved when networks are trained to find other statistical structures in the data.The thesis is organized as follows:Chapter 2: Formulates the recognition task in the framework of the “bias-variance”dilemma. We show that for a good generalization ability the variance portion of thegeneralization error has to be properly controlled. We discuss different methods to controlthe variance portion of the generalization error and present two main approaches: reducingthe variance via ensemble averaging and introducing a learning bias. We review differenttypes of learning bias constraints, and finally, propose reconstruction constraints as anovel type of bias constraints in the context of feed-forward networks. Starting from Section 2.5, we discuss the relation between the “bias-variance” dilemmain statistics, MDL principle and Bayesian framework. We show that the introduction ofa learning bias corresponds to a model-cost in the description length, which has to beminimized along with an error-cost under the MDL principle. At the same time, underthe Bayesian framework, the model-cost corresponds to prior knowledge about the weightsand hidden representation distributions.Chapter 3: Introduces a hybrid feed-forward network architecture, which uses the re-construction constraints as a bias imposing mechanism for the recognition task. This net-work, which can be interpreted under MDL and Bayesian frameworks, modifies the lowdimensional representation by minimizing concurrently the mean squared error (MSE)of reconstruction and classification outputs. In other words, it attempts to improve thequality of the hidden layer representation by imposing a feature selection useful for bothtasks, classification and reconstruction. A significance of each of the tasks is controlledby a trade-off parameter λ, which is interpreted as a hyper-parameter in the Bayesianframework. Finally, this chapter presents technical details about the network architectureand its learning rule.Chapter 4: Discusses various information theory principles as constraints for the clas-sification task. We introduce a hybrid neural network with a hidden representation which
    • Chapter 1: Introduction 6has some useful properties, such as the independence between hidden layer neurons ormaximum information transfer in the hidden layer, etc.Chapter 5: Discusses the face recognition task. We review different Neural Networksmethods used for face recognition and apply the hybrid networks introduced in Chap-ters 3–4. This chapter contains technical details related to face normalization and learn-ing procedures. It is shown that the best regularized network is impractical for degradedimage recognition, and integration over different regularization parameters and differentinitial weights is preferable. This integration is roughly approximated by averaging overnetwork ensembles. We consider three ensemble types: Unconstrained ensemble that cor-responds to integration over initial weights and fixed trade-off parameter λ = 0, i.e. thehidden representation is based on the recognition task alone; Reconstruction ensemblethat corresponds to integration over different values of the trade-off parameter λ for fixedinitial weights. Joined ensemble that corresponds to integration over both the trade-offparameter λ and initial weights and is obtained by merging unconstrained and reconstruc-tion ensembles. Classification results on the degraded images, such as noisy, partially occluded andblurred images are presented. We show that the joined ensemble is superior to the recon-struction ensemble, which in turn is superior to the unconstrained ensemble. Finally weconclude that reconstruction constraints improve generalization, especially under imagedegradations. In addition we show that via saliency maps (Baluja, 1996) reconstructioncan deemphasize degraded regions of the input, thus leading to classification improvementunder “Salt and Pepper” noise.Chapter 6: Addresses recognition of blurred and noisy images. In practice, imagesappear blurred due to motion, weather conditions and camera defocusing. Several meth-ods that address recognition of blurred images are proposed: (i) Expansion the trainingset with Gaussian blurred images; (ii) Constraining reconstruction of blurred images tothe original images during training; (iii) Usage of state of the art restoration methods aspreprocessing to degraded images. Three types of joined ensembles were considered and compared: Ensemble of networkstrained on the original training data only, and ensembles trained on the training setexpanded with Gaussian blurred images and with reconstruction constraints of two types,where the first is a simple duplication of the input in the output and second as describedabove in (ii). It was shown that training with blurred images leads to a robust classification result
    • Chapter 1: Introduction 7under different types of the blur operations and is more important than the restorationmethods.Chapter 7: Summarizes our research and gives some perspective to its future develop-ment, such as: • Testing the hybrid architecture performance on the non face data sets of similar object images, such as military, medical and astronomical • Ensemble interpretation • Using the recurrent network architecture • Weighted network ensemble averaging based on the different error types between input and output reconstruction layers • Using invariance constraints (tangent prop like, see Chapter 2) regularization terms for different types of blur operations for both recognition and reconstruction tasks • Generalization of the proposed hybrid network on the other types of the generative (reconstruction) models constrained by the classification task
    • Chapter 2Statistical formulation of theproblemImages as input to Neural Networks are a very high dimensional data with the size equalto the number of pixels in the image. In this case, the number of the network weightparameters is considerably larger than the size of the training set. This leads to thecurse of dimensionality (Bellman, 1961), which means that there is not enough datato robustly train a classifier in a high dimensional space. Until recently, estimation insuch cases sounded unrealistic, but it is now accepted that such estimation is possibleif the actual dimensionality of the input data is much smaller. In other words, a true,intrinsic dimensionality reduction is possible. A simple dimensionality reduction solelyvia a bottleneck network architecture does not cope with the problem, since a networkcontinues to be an over-parameterized model (i.e. the number of free weight parametersremains large). It is well known that an estimation error is composed of two portions, bias and variance(Geman et al., 1992). The over-parameterized models usually have a small bias (unlessthey are incorrect), but have high variance, since the available data is always small com-pared to the number of the free parameters and this leads to a high sensitivity to noisein the training data. To robustify the estimator, the variance portion of the error hasto be controlled. One of the ways to control variance is via averaging single estimatorstrained on the same task. The other method controls variance by introducing a learningbias as constraints on the network architecture. Different types of smoothing constraintsare widely spread (Wahba, 1990; Murray and Edwards, 1993; Raviv and Intrator, 1996;Munro, 1997). However, as has been pointed out by Geman et al. (Geman et al., 1992)to solve the bias/variance dilemma innovative bias constraints have to be used. Introduc-tion of these constraints into the network model leads naturally to a true dimensionalityreduction (Intrator, 1999). 8
    • Chapter 2: Statistical formulation of the problem 9 Below, we present the bias-variance dilemma and review methods to control the vari-ance and bias portions of the prediction error. Then we propose to use image reconstruc-tion as an innovative bias constraint for image classification. We proceed with discussionon the relation between the “bias-variance” dilemma in statistics, MDL principle andBayesian networks.2.1 Bias-Variance error decomposition for a single predictorThe basic objective of the estimation problem is to find a function fD (x) = f (x; D) given afinite training set D, composed of n input/output pairs, D = {(xµ , yµ )}n µ=1 x ∈ Rd , y ∈R1 , drawn independently according to an unknown distribution P (x, y), which “best”approximates the “target” function y (Geman et al., 1992). Evaluation of the performance of the estimator is usually done via a mean squarederror by taking the expectation with respect to a marginal probability P (y|x):E(x; D) ≡ E[(y − fD (x))2 |x, D] = E[(y − E[y|x])2 |x, D] + E[(fD (x) − E[y|x])2 |x, D] + V ar(y|x) 2E[(y − E[y|x])(fD (x) − E[y|x])|x, D] (2.1.1) =0It can be seen that the third term in the sum is equal to zero, since (fD (x) − E[y|x])does not depend on the distribution P (y|x) and plays the role of a factor, while E[(y −E[y|x])|x, D] is equal to zero. The first term does not depend on the predictor f andmeasures the variability of y given x (in the model with additive independent noise y =f (x) + η(x) this term measures a noise variance in x). The contribution of the secondterm can be reduced by optimizing f . This term measures the squared distance betweenthe estimator fD (x) and the mean of y given x (E[y|x]). A good estimator has to generalize well to new sets drawn from the same distributionP (y, x). A natural measure of the estimator effectiveness is an average error E(x) ≡ED [E(x; D)] = ED [E[(y − fD (x))2 |x, D]] over all possible training sets D of fixed size: E(x) = V ar(y|x) + (ED [fD (x)] − E[y|x])2 + ED [(fD (x) − ED [fD (x)])2 ] (2.1.2) intrinsic error squared bias b2 (f |x) variance var(f |x)The first term is an intrinsic error that can not be altered. If on average, fD (x) isdifferent from E[y|x], then fD (x) is biased. As we can see, an unbiased estimator maystill have a large mean squared error if the variance is large. Thus, either bias or variancecan contribute to poor performance (Geman et al., 1992). When training with a fixed
    • Chapter 2: Statistical formulation of the problem 10training set D, reducing the bias with respect to this set may increase the variance ofthe estimator and contribute to poor generalization performance. This is known as thetradeoff between variance and bias.2.2 Variance control without imposing a learning biasThe variance portion of a prediction error can sometimes be reduced without a bias in-troduction by ensemble averaging. An ensemble (committee) is a combination of singlepredictors trained on the same task. For example, in neural networks, an ensemble is acombination of individual networks that are trained separately and then their predictionsare combined. This combination is done by majority or plurality rules (in classification)(Hansen and Salamon, 1990) or by a weighted linear combination of predictors in regres-sion (Meir, 1994; Naftaly et al., 1997). The plurality rule is defined as the decision agreedby the majority of networks. The majority rule is defined as the decision agreed bymore than half of the networks, otherwise the ensemble rejects to classify and an error isreported. The most general method to create ensemble has been presented by Wolpert(Wolpert, 1992). The method is called stacked generalization and a non-linear networklearns how to combine the network outputs with the weights that vary over the featurespace. It is well known that ensemble is useful if its individual predictors are independentin their errors or disagree on some inputs. Thus, the main question is to find networkcandidates that achieve this independence. One of the widely spread methods to createneural network ensembles is based on the fact that neural networks are non-identifiablemodels, i.e. the selection of the weights is an optimization problem with many localminima. Thus, a network ensemble is created by varying the set of initial random weights(Perrone, 1993). Another way is to use different types of predictors, like a mixture ofnetworks with a different topology and complexity or a mixture of networks with completelydifferent types of learning rules (Jacobs, 1997). Another way is to train the networks ondifferent training sets. Below, a bias-variance error decomposition for a weighed linearcombination of predictors is presented (Raviv, 1998; Tesauro et al., 1995). Let us consider M predictors fi (x, Di ), each trained on a training set Di . All trainingsets have the same size and are drawn from the same joint distribution P (y, x). Considerthe ensemble based on the linear combination of predictors: fens (x) = ai fi (x, Di ), i ai = 1, ai ≥ 0, i = 1, 2, . . . , M. (2.2.3) i
    • Chapter 2: Statistical formulation of the problem 11The normalization condition i ai = 1 is implied to make an ensemble unbiased, wheneach individual estimator fi is unbiased. Let us consider the error (2.1.2) for this ensemble: Eens (x) = V ar(y|x) + b2 (fens |x) + var(fens |x), (2.2.4)where the bias b(fens |x) is given as: b(fens |x) = ED1 ,D2 ,...,DM [ ai fi (x, Di ) − E[y|x]] = i ai EDi [fi (x, Di ) − E[y|x]] = ai b(fi |x). (2.2.5) i iThus the bias of the ensemble is the same linear combination of the biases of the estima-tors. Expanding the ensemble variance term we get: var(fens |x) = ED1 ,D2 ,...,DM [{ ai fi (x, Di ) − ED1 ,D2 ,...,DM [ ai fi (x, Di )]}2 ] = i i ED1 ,D2 ,...,DM [( ai fi (x, Di ) − ai EDi [fi (x, Di )])2 ] = i i ED1 ,D2 ,...,DM [( ai (fi (x, Di ) − EDi [fi (x, Di )])2 ] = i ED1 ,D2 ,...,DM [ a2 (fi (x, Di ) − EDi [fi (x, Di )])2 + i i 2 ai aj (fi (x, Di ) − EDi fi (x, Di ))(fj (x, Dj ) − EDj fj (x, Dj ))] = i>j = a2 var(fi |x) + 2 i ai aj EDi ,Dj [(fi − EDi [fi ])(fj − EDj [fj ])] i i>jFinally, we get the next expression for the ensemble error: Eens (x) = V ar(y|x) + ( ai b(fi |x))2 + a2 var(fi |x) i i i +2 ai aj EDi ,Dj [(fi − EDi [fi ])(fj − EDj [fj ])] (2.2.6) i>jIf all estimators are unbiased, uncorrelated and have identical variances, simple averagingwith the same weights ai = 1/M leads to the following ensemble error (Raviv, 1998): 1 E(x) = V ar(y|x) + b2 (f |x) + var(f |x). MThis decomposition shows that when biases are small and predictors are independent asignificant reduction of order 1/M in the variance may be attained. If estimators are unbiased and uncorrelated it is easy to show that optimal weights 1have to be inversely proportional to the variance of the individual predictors ai ∝ var(fi |x) ,(Tresp and Taniguchi, 1995; Taniguchi and Tresp, 1997). Intuitively it means that apredictor that is uncertain about its own prediction should obtain a smaller weight.
    • Chapter 2: Statistical formulation of the problem 122.3 Variance control by imposing a learning biasA regression function (E[y|x]) is the best estimator. In order to find an unbiased estimator,a family of possible estimators has to be abundant. In the MLP (multi-layer perceptron)networks, this may be attained at the expense of network architecture growing. Thiseliminates bias, but increases variance unless the training data is infinite. In practice, thetraining data is finite and the main question is to make both a bias and variance “small”using finite training sets (Geman et al., 1992). Geman et al. point out that in thislimitation the learning task is to generalize in a very nontrivial sense, since the trainingdata will never “cover” a space of possible inputs. This extrapolation is possible, if theimportant generalizations are prewired in learning algorithms by purposefully introducinga bias. The most general and weakest a-priori constraints assume that mapping is smooth.Other, stronger a-priori constraints may be expressed as an invariance of the mappingto some group of transformation or an assumption about the class of possible mapping.Another type of specific bias constraints appears when a supervised task is learned inparallel with its other related tasks. One way to categorize different types of constraints into two groups: variance andbias constraints, has been proposed in (Intrator, 1999). Both types of constraints serveto reduce the variance portion of the generalization error, however they have a differenteffect on the bias portion of the error. Variance constraints always result in an increase ofthe bias portion of the error. In contrast, bias constraints assist in learning and even mayreduce the bias portion of the error. When networks are learned to satisfy constraints only,the bias constraints lead to a meaningful hidden representation, capturing the structureof the input domain; while a hidden representation extracted via the variance constraintsis less interesting.2.3.1 Smoothness constraintsThe easiest way to smooth the mapping approximated by neural networks is by controllingnetwork structure parameters such as numbers of hidden units and hidden layers. Thelarger is the number of network units, the larger is the number of weight fitting parameters.The over-parameterized models are highly flexible and reduce bias. However, they aresensitive to noise that leads to a large variance and a large generalization error. Anotherway to control smoothness in neural networks, borrowed from the spline theory (Wahba,1990), is to use weight decay. This involves adding a penalty term controlling a weight’s 2norm, to the network cost function E = i yi − f (xi , ω) (other forms of cost functions
    • Chapter 2: Statistical formulation of the problem 13are presented in (Bishop, 1995a)): 2 Eλ = E + λ ω ,where xi and yi are the suitably scaled input and output samples ( z is the normin the space of the element z). Another tightly related approach is to constrain a rangeof the weights to some middle values. The method is called weight elimination and the 2 2 2regularization term has the form λ i ωi /(ωi + ωi0 ). A direct approach is to consider aregularizer which penalizes curvature explicitly: 2 Eλ = E + λ Pf ,where P is a differential operator. Another way to control the smoothness is to inject noiseduring the learning. The noise is usually added to the training data (Bishop, 1995a; Ravivand Intrator, 1996), but may be added to the hidden units (Munro, 1997) or weights(Murray and Edwards, 1993) during learning as well. It has been shown (Bishop, 1995b)that learning with input noise is equivalent to Tikhonov (direct curvature) regularization.Though smoothness constraints bias toward smooth models, they are essentially varianceconstraints.2.3.2 Invariance bias constraintsGiven an infinite training data and unlimited training time, a network can learn theregression function. However, the data is rather limited in practice and this limitationmay be overcome by imposing bias as invariance constraints. One way to implement thisregularization is by training the system with additional data. This data is obtained bydistorting (translating, rotating, etc.) the original patterns (Baird, 1990; Baluja, 1996),while leaving the corresponding targets unchanged. This procedure, called the distortionmodel, has two drawbacks. First, the magnitude of distortion and the number of artificialdegraded patterns have to be defined. Second, the generated data is correlated withthe original training data. This type of regularization is referred to as a data drivenregularization (Raviv, 1998). An alternative way is to impose invariance constraints by adding a regularization termto the mean squared error E (Simard et al., 1992). The regularization term penalizeschanges in the output when the input is transformed under the invariance group. Letx be an input, y = f (x, w) be the input-output function of the network and s(α, x) atransformation parameterized by some parameter α, such that s(0, x) = x. When theinvariance condition for every pattern xµ is written as: f (s(α, xµ ), w) − f (s(0, xµ ), w) = 0 (2.3.7)
    • Chapter 2: Statistical formulation of the problem 14the latter constraint for an infinitesimal α may be rewritten as: ∂f (s(α, xµ ), w) |α=0 = 0, or ∂α ∂s(α, xµ ) fx (xµ , w) · tµ = 0, tµ = |α=0 , (2.3.8) ∂αwhere fx is the Jacobian (matrix) of the estimator f for a pattern xµ , andtµ is a tangentvector associated with the transformation s. The penalty term is written as Ω(f , w) = 2 µ f x · tµ and a penalized function is Eλ = E + λΩ(f , w). This regularization termstates that the function f should have zero derivatives in the directions defined by thegroup of invariance and is called tangent prop. The tangent prop is an infinitesimal form of the invariance ”hint” proposed by Abu-Mostafa (Abu-Mostafa, 1993). The conditions of equivalence between adding distortedexamples and regularized cost function are presented in (Leen, 1995). In particular, it isshown that smoothed regularizers may be obtained as a special case of a random shiftinginvariance group: s(x, α) = x + α, where α is a Gaussian variable with a sphericalcovariance matrix. Obviously, non-trivial invariance constraints belong to a bias type ofconstraints.2.3.3 Specific bias constraintsThese constraints express our a-priori heuristic knowledge about the problem. A com-bination of the Exploratory Projection Pursuit (EPP) method with Projection PursuitRegression (PPR) in feed-forward neural networks (Intrator, 1993a; Intrator et al., 1996;Intrator, 1999) and the multi-task learning (MTL) method (Caruana, 1995), are examplesof this type of the bias constraints.Hybrid EPP/PPR neural networksPPR is a method to perform dimensionality reduction by approximating the desired func-tion as a composition of lower dimensional smooth functions that act on linear dimensionalprojections of the input data (Friedman, 1987). In other words, PPR tries to approximatethe best estimator, that is a regression function f (x) = E[Y |X = x] from observationsD = {(xµ , yµ )}n by a sum of ridge functions gj (functions that are constant along lines): µ=1 m f (x) ≈ gj (aj · x), j = 1, . . . , m. (2.3.9) j=1 In the feed-forward neural networks, the ridge functions are set in advance (as logistic
    • Chapter 2: Statistical formulation of the problem 15sigmoidal, for example) and the output is approximated as m f (x) ≈ βj σ(aj · x), j = 1, . . . , m, x, aj ∈ Rd (2.3.10) j=1where an input vector x is usually extended by adding an additional component equalto 1. Thus, in neural networks only projection directions aj and coefficients βj haveto be estimated. However, when the input is high-dimensional, even the dimensionalityreduction neural networks (m d) are over-parameterized models that require additionalregularization constraints. The already considered smoothness constraint is one way to reduce a variance ofthe network. Another way to impose bias constraints related to the data structure hasbeen proposed by Intrator (Intrator, 1993a). An idea is to train a network (via a back-propagation algorithm) to fit the desired output and to extract a low-dimensional structureof the data using EPP (Friedman, 1987) simultaneously. EPP is an unsupervised methodthat searches in the high dimensional space directions with good clustering properties,characterized by projection indices. An example of combination of supervised learningwith unsupervised using a BCM (Bienestock Cooper and Munro) neuron (Bienenstocket al., 1982; Intrator and Cooper, 1992) has been proposed in (Intrator, 1993b). Thisneuron is learned by minimizing a specific projection index that emphasizes the multi-modality in the data. Computationally, EPP constraints are expressed as minimization of a function ρ(w)measuring the quality of the input after projection and a possible nonlinear transformationφ: ρ(w) ≡ E[H(φ(w · x))], where φ(w · x) is a hidden representation A of the network,H is a function measuring the quality of the hidden representation, and averaging takesplace over an ensemble of the input. The EPP constraints are introduced by modificationa synaptic weight learning rule: ∂wij ∂E(w, x) ∂ρ(w) =− [ + + C], (2.3.11) ∂t ∂wij ∂wijwhere C is an additional complexity penalty term, such as smoothness constraints or thenumber of learning parameters.Multi-task learning (MTL)Another attractive intuitive way to conceive different types of the bias constraints is MTL.MTL is a wide-spread method used in the machine learning. It proposes to learn additionaltasks defined on the same data domain as the special task for improving the generalizationability of the latter. Though the MTL idea is borrowed from the observation that humans
    • Chapter 2: MDL and Bayesian principles 16successfully learn many related tasks at once, it has a rigorous mathematical base. It iseasy to see that the additional task learning in MTL emerges as a bias imposing mecha-nism, that controls the balance between the bias-variance portions of the generalizationerror. The MTL approach in the artificial networks is realized via connectionist networkarchitectures. In connectionist network one shared representation is used for multipletasks. The hidden weights, connected input and this shared representation are updatedas a linear combination of the multi-task gradients in the back propagation of their errors.Such learning moves the shared hidden layer towards representations that better reflectregularities of the input domain. Though the measure of task relation can not be rigorously defined, some mechanismsexplaining the benefit of MTL have been suggested (Caruana, 1995; Abu-Mostafa, 1994).Nevertheless, the way to test the appropriateness of the related task as a proper biasis empirical. It is easy to see that the combination of EPP and PPR neural networkscan be also considered in the MTL framework, though in MTL, a related task is usuallyexpressed more loosely and heuristically than the EPP constraints.2.4 Reconstruction bias constraintsAs shown above in Section 2.3.3, feed-forward Neural Networks which require estimationof many parameters, are subjected to the bias/variance dilemma. We have seen also inSections 2.2–2.3 that different ways to control the bias/variance portion of the predictorerror exist. However, when the dimensionality of the input is very high, innovative waysto reduce the variance portion of the error, as well as methods to impose (reasonable)bias, are required. In this thesis, continuing the previous line of study, we propose a new kind of spe-cific bias constraints for image classification feed-forward networks in the form of theimage reconstruction. We also consider new information theory constraints, seeking di-verse structure in the data and compare the effect of the different constraints on thegeneralization performance of the classification neural network. Below, we discuss Bayesian and minimum description length (MDL) frameworks forlearning in neural networks. We show that the bias-variance dilemma can be naturallyreformulated in the MDL framework, where learning constraints emerge as a model-cost,that has to be minimized along with an error-cost, which is represented as the meansquared error (MSE) on the main learning task.
    • Chapter 2: MDL and Bayesian principles 172.5 Minimum Description Length (MDL) PrincipleIn the MDL formulation, one searches for a model that allows the shortest data encoding,together with a description of the model itself (Rissanen, 1985). One of the first perspec-tives for applying the MDL principle in Neural Networks was pointed out by Nowlan andHinton (1992) for supervised learning. In supervised learning, the output y is predictedfrom the input x which is presented at the input layer. The network model is defined bythe weight parameters. Thus, to specify the desired output y given x, the weights anderrors in the output layer have to be described. If it is assumed that the output errorsare Gaussian, then the number of bits to describe the errors is equal to the mean-squaredrecognition error. The weights are encoded using different weight probability modelsand their descrition length is a negative log of weight probabilities. The weight descrip-tion length is equivalent to different complexity terms and the MDL principle leads to aregularization approach in the Neural Networks. For example, the Gaussian probabilisticmodel leads to the weight decay regularization term (see Section 2.7). A more sophis-ticated form of weight decay is obtained when the weights are encoded as a mixture ofGaussians (Nowlan and Hinton, 1992). Later on the MDL principle was applied for unsupervised learning, in particular forautoencoder networks (Zemel, 1993) (see also Section 3.1.2). The autoencoder networkis a feed-forward network which duplicates the observed input in the output layer. Theautoencoder network has a natural interpretation in the MDL framework (Hinton andZemel, 1994). It discovers an efficient way to communicate data to a receiver. A senderuses a set of input-to-hidden weights and, in general, non-linear activation functions toconvert the input into a compact hidden representation. This representation has to becommunicated to the receiver along with the reconstruction errors and hidden-to-topweights. Receiving the hidden-to-top weights, the receiver reconstructs the input fromthis abstract representation and communicated errors. The description length in this caseconsists of three parts: 1. The set of activities A of the representation units. These are codes that the net assigns to each training input sample. Encoding activities of the representation (hidden) units enables to avoid communication of the hidden weights and does not require the knowledge of the input data X . However, the sender and the receiver have to agree on the a-priori distribution of the internal representation. This part of the message corresponds to the representation-cost. 2. The set of hidden-to-output weights W . This part of the message is represented by the weight-cost.
    • Chapter 2: MDL and Bayesian principles 18 3. The reconstruction error, which is a disagreement between desired and predicted outputs. This part of the message is represented by the reconstruction or the error- cost. In order to evaluate the latter, the sender and receiver have to agree on the probability of the desired output of the network given its actual output. In the standard autoencoder, the weight cost is neglected and the representation costis considered to be small and proportional to the number of network hidden units, since itis assumed that all units participate in the equal parity in the data representation. How-ever, instead of the direct evaluation of the representation code, the autoencoder with abottleneck in the hidden layer is trained to minimize the MSE reconstruction error. Incontrast, in the nonstandard versions of autoencoders (Zemel, 1993), the representationcost is evaluated explicitly and its minimization encourages sparse distributed represen-tation, where only few neurons are active, which are responsible for the presence of thespecific features in the patterns. The main difference between the MDL principle for supervised and unsupervised learn-ing proposed by Zemel may be understood considering the unlimited number of trainingsamples. When the number of patterns is infinite, the model cost of the supervisedlearning, which is the cost of the weights, is negligent. In contrast, in the unsupervisedlearning, the model cost never vanishes and the MDL is applied per sample to minimizerepresentation cost and to maintain data fidelity. In this thesis, we combine supervised and unsupervised learning in the hybrid re-construction/recognition network and formulate the MDL principle for this case (see Sec-tion 3.2.3). It turns out that this interpretation is three-fold, depending on what is definedas the main task: 1. When the main task is reconstruction (Gluck and Myers, 1993, a hippocampus model), the reconstruction MSE is an error cost and the recognition MSE is a model cost (or a representation cost, since the MSE recognition error depends on the hidden layer representation and the recognition top weights that must not affect on the description length). Thus, the network maintains the data fidelity and encourages representation with a good discriminative property. 2. When the main task is recognition and it is assumed that the sender observes both the input and output, while the receiver sees only the input, the recognition MSE is an error cost as in supervised learning and the reconstruction MSE is a model cost (or a representation cost). However, in contrast to a standard supervised learning the representation cost never vanishes.
    • Chapter 2: MDL and Bayesian principles 19 3. When the main task is recognition, but the receiver does not see both x and y, he has in parallel to reconstruct x and predict y. Thus, the sender encodes x, taking into account also the dependence of y on x. He sends the encoded data and errors of recognition and reconstruction outputs, since in the supervised learning the task is to predict y for the given x. In this case, both the recognition and reconstruction MSE stand for error codes and the representation cost is restricted to a small number of the hidden units.2.5.1 Minimum description lengthMDL can be formulated based on an imaginary communication game, in which a senderobserves the data D and communicates it to the receiver. Having observed the data, thesender discovers that the data has some regularity that can be captured by a model M.This fact encourages the sender to encode the data using a model, instead of sending thedata as it is. Due to noise, there are always aspects of the data which are unpredicted bythe model, that can be seen as errors. Both the errors and the model have to be conveyedto the receiver to enable him to reproduce the data. The goal of the sender is to encodedata so that it can be transmitted as accurately and compactly as possible. It is clear, that complex models allow to achieve a high accuracy, but their descriptionis expensive. In contrast, models which are too simple or wrong, are not able to extract thedata regularity. Intuitively, such a communication game can be thought of as a tradeoffbetween the compactness of the model and its accuracy. To transmit the data the sender composes a message consisting of two parts. The firstpart of the message with a length L(M ) specifies the model and the second with a lengthL(D|M ) describes the data D with respect to the model M. The goal of the sender isto find a model that minimizes the length of this encoded message L(M, D), called thedescription length: L(M, D) = L(D|M ) + L(M ), (2.5.12) According to Shannon’s theory (Shannon, 1948; Cover and Thomas, 1991) to encodea random variable X with the known distribution p(X) by the minimum number of bits,a realization x has to be encoded by − log p(x) bits. Thus the description length (2.5.12)is represented as: L(M, D) = (− log p(D|M ) − log p(M )), (2.5.13)where p(D|M ) is the probability of the output data given the model, and p(M ) is ana-priori model probability. The MDL principle requires searching for a model M that
    • Chapter 2: MDL and Bayesian principles 20minimizes the description length (2.5.13): M = arg min(− log p(D|M ) − log p(M )). (2.5.14) M As we have seen in Section 2.1, in the supervised learning the problem is to find a modelthat describes output y as a function of input x based on the available input/output pairsD = {(xµ , yµ )}n . In a standard application of MDL to supervised learning, the output y µ=1is treated as the data D that has to be communicated between the sender and the receiver,while the input data X is assumed to be known by them. Therefore, all the probabilitiesin the formula (2.5.13) are conditioned on the input data, i.e. p(M ) ≡ p(M |X ) andp(D|M ) ≡ p(D|M, X ). However, to simplify the notation we omit X in these expressions. The connection between MDL and Bayesian theory for Neural Networks is demon-strated in the next section.2.6 Bayesian frameworkIn the Bayesian framework, one seeks a model that maximizes a posterior probability ofthe model M given the observed input/output data (X , D): p(D|M, X )p(M |X ) p(M |D, X ) = , (2.6.15) p(D|X )Usually, in the feed-forward networks trained by supervised learning the distribution ofthe input data p(x) is not modeled1 . Thus, in (2.6.15), X always appears as a conditioningvariable, which we omit to simplify the notation (similar to the convention accepted forthe description length evaluation): p(D|M )p(M ) p(M |D) = . (2.6.16) p(D) Since p(D) does not depend on the model and the most plausible model M has tominimize the negative logarithm of the posterior probability, we get: M = arg min[− log(p(D|M )) − log(p(M ))]. (2.6.17) M Usually, to apply both the MDL and Bayesian frameworks, one decides in advance on aclass of parameterized models and then searches within this class of parameters to optimizea corresponding criterion. The probability of the data, given a model parameterized byw, can be computed by integrating over the model parameter distribution: p(D|M ) = p(D|M, w)p(w|M )dw. (2.6.18) 1 In Section 3.2.3 we will consider the effect of such modelling.
    • Chapter 2: MDL and Bayesian principles 21Using the Bayesian formula we get: p(w, D|M ) p(D|M, w)p(w|M ) p(w|M, D) = = , (2.6.19) p(D|M ) p(D|M )that shows that a posterior probability of the weights p(w|M, D) is proportional top(D|M, w)p(w|M ). It is usually assumed that a posterior probability of the weightsp(w|M, D) is highly peaked at the most plausible parameter w , and the integral (2.6.18)may be approximated by the height of the peak of the integrand p(D|M, w)p(w|M ), timesa width of this distribution ∆w|M,D (MacKay, 1992): p(D|M ) ≈ p(D|w , M ) × p(w |M )∆w|M,D (2.6.20) best f it likelihood Occam f actorThe quantity ∆w|M,D is the posterior uncertainty in w. Assuming that the prior p(w |M )is uniform on some large interval ∆0 w, representing the range of values of w that the 1model M admits before seeing the data D, p(w |M ) simplifies to p(w |M ) ≈ ∆0 w , and ∆w Occam f actor = . (2.6.21) ∆0 wThus the Occam factor is the ratio of the posterior accessible volume of the model pa-rameter space to the prior accessible volume. Typically, a complex model with manyparameters, has larger prior weights uncertainty ∆0 w. Thus, the Occam factor is smallerand it penalizes the complex model more strongly (MacKay, 1992). Another interpretation of the Occam factor is obtained by viewing the model M ascomposed of a certain number of equivalent sub-models. When data arrive, only onesub model survives and thus the Occam f actor appears to be inversely proportional tothe number of sub models. Thus, − log(Occam f actor) is the maximal number of bitsrequired to describe/indicate this remaining sub model. Using the Occam factor (2.6.21) the condition (2.6.17) states that the most plausiblemodel has to minimize the description length: L(M, D) = − log p(D|w , M ) − log p(M ) − log(Occam f actor) (2.6.22) inaccuracy f or the best parameters model complexityThe first term in (2.6.22) is the ideal shortest message that encodes the data D usingw and characterizes inaccuracy of the model prediction for the best parameters. Thesecond term characterizes the complexity of the model. The more complex the modelis, the less is the discrepancy between the data and their prediction, but this accuracyis achieved at the expense of the model description. This relationship between a modelaccuracy and complexity is tightly related to the bias-variance dilemma considered in
    • Chapter 2: MDL and Bayesian principles 22the previous section. We have seen that the introduction of many parameters leads to abetter accuracy (decreases bias), but incurs high variance. Thus MDL and the Bayesianapproach offer the natural way to resolve the dilemma by seeking a model with a goodgeneralization ability. Another MDL interpretation to (2.6.20) is straightforward: L(D, M ) = − log p(D|w , M ) − log p(w |M ) − log ∆w|M,D − log p(M ). (2.6.23) error−cost weight−cost precision−costThe first term in (2.6.23) is the length of the ideal shortest message that encodes thedata D using the best parameters w . The second term is the number of bits requiredto encode the best model parameters. In addition, the negative logarithm of uncertaintyabout parameters after observing the data (− log ∆w|M,D ) penalizes models which haveto be described with a high precision to fit the data. Usually, the third component isneglected since model parameters are communicated only once, while the data arrive oneafter another. A way to take the third component into consideration in neural networks,but neglecting the second term, describing the a-priori knowledge about the model pa-rameters, has been considered in (Hochreiter and Schmidhuber, 1997).2.7 MDL in the feed-forward NNA feed-forward neural network is an example of the parameterized models that is rep-resented graphically as a feed-forward diagram of several layers of activation units, con-nected by the so called synaptic weights that represent the model parameters. The neuralnetwork architecture allows to evaluate the output data as a function of the input data.The network is supplied by the input data presented in the low input layer of the network.The input is successively propagated via the hidden layers using the weights and networkunits’ activation functions in the forward direction to get the output data D in the topoutput layer of the network. The network weights, the number of hidden units and theactivation unit functions are the main parameters that define the network complexity. Ingeneral, it is often assumed that the network architecture is already defined and the mainproblem is to find the weight parameters. Implementing the MDL principle in neural networks is easy. For simplicity we considertraining a single hidden layer feed-forward neural network (Figure 2.1). Neglecting thethird term in the description length (2.6.23) and assuming that the models have the same
    • Chapter 2: MDL and Bayesian principles 23 Supervised feed-forward network Output W - top weights Hidden representation - A w - hidden weights Input - XFigure 2.1: Feed-forward supervised network. A single arrow between two layers indicatesthat the units of both layers are fully connected.a-priori probabilities p(M ) an optimal weight vector has to minimize 2 : L(M, D) = − log p(D|w, W, M ) − log p(w, W|M ) +const (2.7.24) error−cost model−costThe first term in this expression is the error-cost of specifying the data for the givenweights, i.e. the cost of specifying the errors between true and predicted by the modelswith the given weights outputs. The second term is the model-cost. To evaluate the error-cost, the receiver and the sender have to agree on the specificform of the conditional distribution of the output t ∈ Rn . In the assumption of theindependent Gaussian additive noise with zero mean in the output layer, the posteriorprobability of the output is given by: 1 λ ˆ(x, w, W) − t 2 p(t|x, w, W) = exp(− t ), (2.7.25) C n (λ) 2 2πwhere C(λ) = λ and the parameter λ is inversely proportional to the Gaussian variance 2(λ = 1/σ ). Provided the samples are drawn independently from the distributions (2.7.25) we get: r p(D|w, W, M ) = p(ti |xi , w, W), (2.7.26) i=1 2 We have omitted the super-index for convenience
    • Chapter 2: MDL and Bayesian principles 24where r is the number of training samples. The assumptions (2.7.25) and (2.7.26) produce 1 λ p(D|w, W, M) = nr exp(− ED ), where C (λ) 2 r ED = ˆ(xi , w, W) − ti t 2 . (2.7.27) i=1 When the weight probability distribution is Gaussian and the hidden w and topweights W are independent we get: p(w, W|M ) = p(w|M )p(W|M ) 1 γw p(w|M ) = Nw exp(− w − mw 2 ), C (γw ) 2 1 γW p(W|M ) = N exp(− W − mW 2 ), (2.7.28) C W (γW ) 2where Nw , NW are numbers of the hidden and top weights, coefficients γw , γW are inverselyproportional to the corresponding Gaussian variances and mw , mW are mean values of thehidden and top weights, respectively. Assumptions (2.7.25,2.7.28) lead to the followingexpression for the description length (2.7.24): λ γw 2 γW 2 L(M, D) = ED + w − mw + W − mW + 2 2 2 error weight decay Nw log C(γw ) + NW log C(γW ) + nr log C(λ) + const (2.7.29)The first term may be recognized as an error and the next as a modified weight decayterm. The third term is constant for a chosen net architecture. Thus, the weight-decayterm controls a network complexity imposing smoothness constraints. Another form ofweight decay term has been obtained by modelling the weights as a mixture of Gaussians(Nowlan and Hinton, 1992). There is a deep relationship linking the MDL approach and regularization techniques.The intuitive idea is that complex models can fit better training data, but are not robustto small variations in the data. This relationship between a generalization ability of themodel and its complexity is related to the bias-variance dilemma in statistics (Gemanet al., 1992): over-parameterized models have high variance, while restricting the modelparameters incurs a high bias in the generalization error. The MDL formulation allowsto control bias and variance in a natural way.2.7.1 MDL and EPP bias constraintsLet us assume again that a network architecture, such as a number of hidden units andnonlinear activation functions, is fixed. Nevertheless, does there exists another way to
    • Chapter 2: MDL and Bayesian principles 25control complexity of the network? It turns out that this can be done by imposing biasconstraints on the supervised neural network. A general framework for imposing EPPbias constraints in neural networks (Figure 2.2) has been considered in Section 2.3.3.We have seen that computationally these constraints are expressed as a minimization Hybrid network with EPP constraints Output W - top weights Hidden Bias constraints representation - A w - hidden weights Input - XFigure 2.2: A hybrid feed-forward network with exploratory projection pursuit (EPP)constraints. A single arrow between two layers indicates that the units of both layers arefully connected.of some function H, measuring the quality of the hidden layer representation A, andaveraged over an ensemble of the input. In other words, EPP constraints are constraintson the specific form of the hidden representation that are known a-priori. Thus, theprojection index ρ(w) is a complex function depending on the hidden weights via thehidden representation A: ρ(w) ≡ E[H(A)], where A = f (w, x) and H measures thequality of the hidden representation. This form of constraints may be easily wired inthe MDL framework assuming a particular form of a-priori probabilities of the hiddenweights: µ p(w|M ) = CH (µ) exp(− E[H(f (w, x))]), (2.7.30) 2where CH (µ) is a normalization constant. The a-priori probability p(w|M ) (2.7.30) doesnot depend on the input x explicitly, although it does, since in the Bayesian formulation(2.6.16) all the probabilities have to be conditioned by the input data X . Assuming
    • Chapter 2: MDL and Bayesian principles 26independence of the hidden and top weights, we get: 1 1 L(M, D) = λED + µE[H(A)] − log p(W|M ) +const. (2.7.31) 2 2 weight−cost error−cost representation−costThe expression for the description length (2.7.31) gives a deeper level of description tothe data communication and is close (though not equivalent) to Zemel’s interpretation ofMDL (Zemel, 1993). In Zemel’s interpretation one gets a more realistic interpretation of the communicationgame, where a real communication takes place between the hidden layer with internalrepresentation A and the top layer. The receiver requires three items in order to be ableto recover the desired output: 1. The set of activities A of the representation units; these are codes that nets assign to each training input sample. Encoding activities of the representation (hidden) units avoids communication of the hidden weights and does not require the knowledge of the input data X . However, the sender and the receiver have to agree on the a-priori distribution of the internal representation. This part of the message corresponds to the representation-cost. 2. The set of hidden-to-output weights W . This part of the message is represented by the weight-cost. 3. Reconstruction error, which is a misfit between desired and predicted outputs. This part of the message is represented by the reconstruction or the error-cost. In order to evaluate the latter, the sender and receiver have to agree on the probability of the desired output of the network given its actual output.Usually, the weight-cost, i.e. the number of bits required to communicate the hidden-to-top weights, is not taken into account, since it has to be communicated only once,while representation-cost and error-cost have to be sent for every sample. Thus, the maincommunication tradeoff takes place between representation and error costs. Reducingdimensionality of the data in the hidden layer, i.e. compressing the data, a shorterdescription is obtained, but at the same time the errors are larger. The MDL principle isa tool for achieving a good data representation that is compact and accurate. We see that similar to Zemel’s interpretation of MDL, imposing EPP constraints leadsto the description length (Eq. 2.7.31) that consists of three parts. It requires the sameagreement on probabilities of hidden representation and errors between the sender andreceiver as described above. However, the representation cost in (Eq. 2.7.31) is taken only
    • Chapter 2: MDL and Bayesian principles 27once for all samples, while in Zemel’s interpretation it is permanent and is assigned to eachtraining input sample. When the number of input patterns is infinite, the representationcost induced by EPP constraints is negligible. Thus, in a manner similar to supervisedlearning, EPP constraints lead to a model in which model cost vanishes as the number ofinput patterns becomes infinite. We postpone the consideration of the hybrid autoencoder network with reconstructionconstraints and its MDL interpretation to the next section, where reconstruction task andits application are considered.
    • Chapter 2: Regularization problem 282.8 Appendix to Chapter 2: Regularization problemRegularization may be expressed as a minimization problem with a goal function that isa penalized cost function: 2 Eλ = E + λΩ(f , w), E= yi − f (xi , ω) . iA large value of the regularization parameter λ leads to a network with a large bias(unless the regularization term captures the underlying structure of the data), while asmall value reduces bias but increases variance. Then the regularization task is to findan optimal parameter λ and corresponding model parameters ωλ providing the minimalgeneralization error: 2 Eλ = E[ y − f (x, ωλ ) ].This task is computationally very expensive.Split-sample validation and hold-out method The simplest way to find the regu-larization parameter is to use split-sample validation. This process includes the followingsteps for each tested value of the regularization parameter λ (this process is common forthe choice of the other regularization parameters, such as the number of hidden units, achoice of the early time stopping moment, etc.): • A random data is split into a training and validation set. Often 2/3 of the data is used for training and 1/3 for testing. • The training set is used for estimation of the predictor parameters by minimizing Eλ . • The validation set is used to test a prediction error (E). The validation set must not be used in any way during training. • The predictor with the smallest prediction error corresponds to the optimal regu- larization parameter λ.The generalization error of the best predictor is in general too optimistic. The predictionerror on a third separately kept data set, called the test set is more realistic and isoften reported as the result of the predictor accuracy. This method is called the hold-outmethod. The disadvantage of the split-sample validation and hold-out method is that theyreduce the amount of data available for both training and validation. Two methods that
    • Chapter 2: Regularization problem 29overcome this drawback are cross-validation and bootstrapping (Efron and Tibshirani,1993; Bishop, 1995a).Cross-validation In k-fold cross-validation, the data is divided into k subsets of (ap-proximately) equal size. A network is trained k times, each time leaving out one of thesubsets from the training set and using the omitted subset as a validation set to computean error. If k equals the sample set size, this is called “leave-one-out” cross-validation.“Leave-v-out” is a more elaborate and expensive version of cross-validation that involvesleaving out all possible subsets of v cases. A generalization error is then measured as anaverage performance over all possible validation tests. Cross-validation is an improvementon split-sample validation.Bootstrapping In many cases, bootstrap seems to be better than cross-validation(Efron and Tibshirani, 1993). In the simplest form of bootstrapping, the training data isbootstrapped, instead of repeatedly analyzing subsets of the data as in cross-validation.Given a data set of size n, a bootstrap sample is created by sampling n instances uni-formly from the data with replacement. Then the probability of the instance to remain inthe test set is (1 − 1/n)n ≈ e−1 ≈ 0.368; and to be in the training data is 0.632. Given anumber b of bootstrap samples, the average performance is evaluated as a weighted sumsof the training (Eitraining ) and testing (Eitesting ) errors: 1 b E= (0.632Eitraining + 0.368Eitesting ) (2.8.32) b i=1Usually the number of recommended bootstrap samples is between 200 − 2000 (Kohavi,1995). Cross-validation and bootstrapping require many runs that may be computationallyprohibitive, especially for the most interesting perception tasks, when the input dimension-ality is very high. Both cross-validation and bootstrapping work well for continuous errorfunctions, such as the mean squared error, but it may perform poorly for non-continuouserror functions, such as the misclassification rate.
    • Chapter 3Imposing bias via reconstructionconstraints3.1 IntroductionReconstruction is one of the important tasks of the complex visual processing. It isa process of reproducing the input via some reasonably well chosen model. It is com-monly assumed that there is a compression via a bottleneck model and thus, the inputis reproduced from a reduced internal representation. The oldest and widely spread re-construction method is Principal Component Analysis (PCA). PCA is an optimal linearcompression, that is based on minimization of the mean squared error between input andits reconstruction. A simple generalization of PCA, in the nonlinear case, is a nonlinearautoencoder. Below, we present both these models and discuss their relationship to theMDL principle. We proceed then with a more general notion of reconstruction via a generative modeland reexamine diverse applications of the reconstruction models. Finally, we introduce anovel method that uses reconstruction as a bias constraint to a supervised classificationtask.3.1.1 Principal Component Analysis (PCA)PCA is widely used in multivariate analysis (Duda and Hart, 1973). PCA, also known asthe Karhunen-Lo´ve transformation (Oja, 1982; Fukunaga, 1990), is a process of mapping ethe original data into a more efficient representation, using an orthonormal linear trans-formation that minimizes the mean squared error between the data and its reconstructedversion. It is well-known that the optimal orthogonal basis of the data space is formed by theeigenvectors of the covariance matrix of the data. New data representation is obtained 30
    • Chapter 3: Reconstruction constraints 31by projecting the data to this new optimal basis. The eigenvectors corresponding to thelargest eigenvalues are the most significant (accounting for most of the variance in thedata). Thus, discarding coordinates in these directions, leads to the largest error in themean-squared sense. Therefore, the coordinates corresponding to the small eigenvaluesshould be deleted first, when compression is performed. Different PCA algorithms using neural networks have been reported (Haykin, 1994,see review). The first PCA network proposed by Oja (1982), uses a Hebbian learningrule to find the first eigenvector corresponding to the maximal eigenvalue. It’s gener-alized version, called the generalized Hebbian network (GHA) (Sanger, 1989), extractsthe first successive eigenvectors and uses feed-forward connections only. A modificationof GHA, an adaptive principal component extraction (APEX) algorithm (Kung and Dia-mantaras, 1990), uses additional lateral connections to decorrelate network outputs. GHAand APEX are examples of reestimation and decorrelating types of the PCA algorithms,respectively. PCA using Hebbian networks has been considered as a first principle of perceptualprocessing (Miller, 1995; Atick and Redlich, 1992; Hancock et al., 1992; Field, 1994). Themain goal of these studies is to explore the similarities between the PCA eigenvectorsand the receptive fields of cells in the visual pathway. It may be shown (Fukunaga, 1990;Gonzalez and Wintz, 1993; Field, 1994), that for stationary and ergodic processes, PCA isapproximately equivalent to the Fourier transform. The natural images are not stationary,however, and their covariance matrix does not describe completely the data distribution.It has been recently shown (Hancock et al., 1992), that the first 3−4 eigenvectors extractedfrom Gaussian smoothed natural images resemble ”Gabor functions”, that provide goodmodels of cortical receptive fields. However, the following eigenvectors no longer look likecortical receptive fields. PCA extracts a fully distributed representation, because onlyfew neurons that carry most of the variance are kept, and thus all components of theobservation vector participate in its projection into the eigenspace. Below, we present autoencoder network that is tightly related to PCA and discuss itsinterpretation in the MDL framework.3.1.2 Autoencoder network and MDLAn autoencoder network (Figure 3.1) is a feed-forward multi-layer perceptron (MLP) net-work with the output layer coinciding with the input layer. Usually, it contains a singlehidden layer, though variants with additional hidden layers have been also considered(Kramer, 1991). The number of the hidden units is assumed to be much less than dimen-sionality of the input. Therefore, it reduces dimensionality of the input extracting the
    • Chapter 3: Reconstruction constraints 32 Autoencoder network architecture W - hidden-to-top weights w - hidden weights Figure 3.1: Reconstruction of the inputs is done from the hidden layer representation.so-called internal representation in the hidden layer. The autoencoder network has a natural interpretation in the MDL framework (Hintonand Zemel, 1994). It discovers an efficient way to communicate data to a receiver. Asender uses a set of input-to-hidden weights and, in general, non-linear activation functionsto convert the input into a compact hidden representation. This representation has tobe communicated to the receiver along with the reconstruction errors and hidden-to-topweights. Knowing the hidden-to-top weights the receiver reconstructs the input from thisabstract representation and communicated errors. From Eq. 2.7.24 the description length is composed of the error-cost and the model-cost. Assuming that the errors are encoded using a zero-mean Gaussian with the samepredetermined variance for each output unit, the error-cost is given by the sum of thesquared errors. Since in the autoencoder the hidden units are always active, the model costmay be approximated by the size of the hidden layer. Often, the model cost is ignored,and the MDL principle leads to a simple minimization of the sum of squared errorsvia a network with a bottleneck structure. Thus, the autoencoder learns the compactrepresentation of the input. In addition, the bottleneck structure forces the network tolearn prominent features of the input distribution which are useful for generalization. Thenetwork is robust to noise and may be used for pattern completion, when part of the inputis corrupted or absent. A linear one-hidden layer autoencoder is closely related to PCA, since its hiddenweights span the same subspace as found by principal eigenvectors (Bourlard and Kamp,1988). However, contrary to PCA, the hidden weights are not forced to be orthogo-nal and do not coincide with the hidden-to-top weights. The analytical solution of the
    • Chapter 3: Reconstruction constraints 33optimization problem imposed by the linear autoencoder is given by: W = UT−1 , w = TUt (3.1.1)where T ∈ Rp×p is an arbitrary nonsingular scaling matrix; U ∈ Rn×p (p ≤ n) is amatrix of the principal eigenvectors stacked by columns; W and w are hidden-to-topand hidden weights respectively; n and p are the number of units in the input and hiddenlayers respectively. However, since learning in the autoencoder relies on a gradient descenttechnique it can get trapped in local minima. In the nonlinear case, Bourlard and Kamp claim that nonlinear and linear autoen-coders are equivalent, since when the norm of the scaling matrix T is infinitely small,sigmoidal activation functions can be approximated arbitrary close by linear activationfunctions. However, their proof is valid only from the reconstruction error minimizationviewpoint, and not the extracted internal representation context. Their analysis doesnot take into account a convergence issue. Indeed, to make nonlinear and linear autoen-coder solutions arbitrarily close, the norm of the matrix T has to be arbitrarily small (forexample, by introducing some scaling parameter → 0). While is positive the linearautoencoder hidden weights span the same space as the principal eigenvectors, but atthe same time there is a difference between hidden weights extracted by the linear andnonlinear autoencoders. This difference disappears only for = 0, when the matrix Tbecomes singular. Thus, it is not obvious that the hidden weights obtained in the limitof this convergence span the space extracted by the principal eigenvectors. It has been recently shown, that when the data is whitened (i.e. the data covariancematrix is unit and spherical) and non-linear activation functions are adjusted properly, theautoencoder is able to extract the independent components (Oja, 1995a) (i.e. responsesof different hidden neurons are independent, see also Chapter 4), while the PCA solutionis not well defined. Thus, the non-linear autoencoder can be made sensitive to higherorder statistics, while PCA is sensitive to the second order statistics of the data. The presence of the proper nonlinearities in the autoencoder allows to extract sparserepresentation, while PCA forms distributed representation. In the distributed represen-tation, all the hidden units participate in the pattern encoding, while in the sparse, onlya few are active, which are responsible for the presence of some specific features in thepattern. PCA forms the distributed representation, since only few neurons which carrymost of the variance are kept for data reconstruction and they are active for all patterns. Other variants of autoencoders that encourage sparse hidden representations havebeen proposed by Zemel (1993). The code-cost of the sparse representation is small, evenwhen the number of hidden units is large. Thus, though these autoencoders are trained
    • Chapter 3: Reconstruction constraints 34to minimize a sum of the representation (code) and error costs, they do not necessaryhave a bottleneck structure and develop interesting biologically plausible representations.3.1.3 Reconstruction and generative modelsThere is evidence in several psychological experiments (for example, completion of par-tially occluded contours (Lesher, 1995)) that humans perceive a reconstructed versionof the input instead of the raw ambiguous input. The reconstruction may be a morecomplex process than simple duplication of the incoming information, including deblur,denoising, completion of occluded areas, etc. It is often assumed that the observed signalsare synthesized by some generative model from an abstract internal representation. Thus,the reconstruction is considered to be composed of two phases (Hinton and Ghahramani,1997). The first phase is a recognition phase, inferring the underlying internal repre-sentation of the incoming input and the second – a generative phase converts internalrepresentation into an input form (reconstructed object). From a statistical viewpoint, learning to reconstruct is the problem of maximizingthe likelihood of the observed data under a generative model. This estimation is oftenan ill-posed problem, that can be solved using the expectation maximization (EM) algo-rithm (Dempster et al., 1977; Neal and Hinton, 1993). This iterative algorithm increases(or does not change) maximum likelihood in every iteration, which consists of two steps,expectation and maximization. In EM, the recognition phase corresponds to the expec-tation step (E-step) and generative phase to the maximization (M-step). In the E-step, adistribution of the internal representation is estimated from the observed data and currentmodel parameters. Using this distribution and the observed data, the generative modelparameters are updated via an average likelihood maximization. Different generative models and assumptions about distribution of the internal rep-resentation lead to different network models and sensory representations. The inferencephase is difficult. In logistic belief networks (LBN) and Boltzmann machine (Hinton andGhahramani, 1997) the hidden state is picked using Gibbs sampling, i.e. each unit is vis-ited one at a time and its new state is stochastically picked from its posterior distributiongiven the current states of all the other units (Jordan, 1999, comprehensive survey). Inthe wake-sleep algorithm (Hinton et al., 1995), a model uses separate bottom-up recogni-tion connections to pick up binary states for units in one layer, given the already selectedbinary states of units in the layer below. Both PCA and the autoencoder network may be interpreted as generative models.PCA as a generative model emerges as a constrained case of factor analysis (Roweis andGhahramani, 1997; Hinton and Ghahramani, 1997). In factor analysis the observation
    • Chapter 3: Reconstruction constraints 35is a linear transformation of the hidden variables, corrupted with an additive sensorynoise that is Gaussian. The linear transformation is realized via a matrix of the gener-ative weight vectors. Each generative weight vector connects hidden variables with thecorresponding observation variable. Hidden variables are referred to as factors and areassumed to be Gaussian. PCA is obtained when the covariance matrix of the sensorynoise is assumed to be a scaled identity matrix I, with the infinitesimal scaling factor → 0. In this limiting case, the posterior distribution of the hidden variables shrinks to asingle point, i.e. given the observation, the hidden representation becomes non random.In PCA the generative weight vectors are forced to be orthogonal, that leads to a simplerecognition of the deterministic hidden representation as a linear transformation with thematrix of the recognition weight vectors equal to the transpose of the generative weightmatrix. Interpretation of PCA as a generative model disregards the order of the hidden vari-ables, but allows the use of EM for the extraction of eigenvectors (Roweis, 1997). Thismethod is especially efficient for high dimensional data, where a covariance matrix is notfull rank and has a large size that makes the simple diagonalization of the covariancematrix computationally difficult. The transformation from the input to the hidden layer in the autoencoder net isassociated with the recognition phase and from the hidden layer to the output as thegenerative phase. Therefore, the hidden weights emerge as recognition weights and thehidden-to-top weights as generative weights.3.1.4 Classification via reconstructionAs we have shown above, an implicit reconstruction goal is to find a meaningful internalrepresentation of the data that can be obviously used for data compression and commu-nication. Interpreted as a set of good features, it may be applied for further processingand learning. This usage is not absolutely apparent, since during feature extraction someinformation is lost. Below, we consider some examples of using internal representationsextracted via reconstruction for recognition.PCA for classification PCA was first used as a means of preprocessing for subsequentface recognition in (Kirby and Sirovich, 1990; Turk and Pentland, 1991). Later PCA wasused for a man-made object recognition and pose estimation (Murase and Nayar, 1993). PCA proceeds by scanning and representing images as points of a high dimensionalspace with the dimension equal to the number of image pixels. The eigenvectors of thedata covariance matrix represented as images are called the eigenpictures. The first large
    • Chapter 3: Reconstruction constraints 36eigenvectors form the basis of a low-dimensional subspace, called the eigenspace. All thesample-images and new images of the objects are projected into the eigenspace and therecognition problem is solved in the reduced dimensional space by different statisticalmethods (nearest neighbor rule, vector quantization, etc.). Though application of PCA for recognition has been relatively successful, a questionof the PCA optimality for recognition task has been also addressed (Turk and Pentland,1993; O’Toole et al., 1993). Experimental studies of Turk et al. (1993) show that the firstfew eigenfaces primarily capture the effects of changing illumination and neglecting thefirst few eigenfaces can lead to a substantial increase in the recognition accuracy. Thisobservation has been supported by a different study (O’Toole et al., 1991; O’Toole et al.,1993). It has been shown that a low-dimensional representation of the faces associatedwith the small eigenvalues is better for face classification and familiarity, than a high-dimensional representation associated with the large eigenvalues when these spaces havethe same small dimensionality. The explanation of a PCA utility is based on the fact that the eigenvectors correspond-ing to the large eigenvalues are the directions with the large data variability (Figure 3.2a).Thus, it seems reasonable that these directions are good for recognition. However, thisassumption fails as can be easily seen from Figure 3.2b. This figure demonstrates the main Eigenspaces extracted by PCA a b 1 0 11 00 11 1 00 0 11 00 11 11 00 00 11 1 1 1 00 0 0 0 11 11 00 00 11 11 1 1 00 00 0 0 11 1 1 1 1 1 00 0 0 0 0 0 11 11 00 00 1 0 11 11 1 11 1 00 00 0 00 0 11 1 1 1 1 1 1 1 00 0 0 0 0 0 0 0 11 11 111 1 00 00 000 0 11 11 1 1 1 1 1 1 00 00 0 0 0 0 0 0 11 1 1 1 1 1 1 1 1 1 00 0 0 0 0 0 0 0 0 0 11 11 11 111 1 00 00 00 000 0 11 11 10 11 0 0 0 0 0 e1 0 0 1 0 1 0 1 11 1 00 0 11 11 1 11 1 11 11 1 00 00 0 00 0 00 00 0 11 1 1 1 1 1 1 1 1 1 1 1 1 00 0 0 0 0 0 0 0 0 0 0 0 0 11 11 00 00 11 00 11 11 00 00 e1 11 11 1 11 1 11 11 1 11 00 00 0 00 0 00 00 0 00 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1111 11 111 111 11 111 0000 00 000 000 00 000 11 11 111 111 11 111 00 00 000 000 00 000111 11 11 111 111 11 111000 00 00 000 000 00 000111 11 11 111 111 11 111000 00 00 000 000 00 000 111 11 11 111 111 11 111 000 00 00 000 000 00 000 111 11 11 111 111 11 111 000 00 00 000 000 00 000 111 11 11 111 111 11 111 000 00 00 000 000 00 000 111 11 11 111 111 11 111 000 00 00 000 000 00 000 111 11 11 111 111 11 111 000 00 00 000 000 00 000 111 11 11 111 111 11 111 000 00 00 000 000 00 000 111 11 11 111 111 11 111 000 00 00 000 000 00 000 111 11 11 111 111 11 111 000 00 00 000 000 00 000 111 11 10 111 111 11 111 0 0 0 0 0 0e2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 01 0 0 0 0 0 0 0 0 0 0 0 111 11 10 111 111 11 111 0 0 0 0 0 01 0 0 0 0 0 0 0 0 0 0 0 111 11 10 111 111 11 111 0 0 0 0 0 01 0 0 0 0 0 0 0 0 0 0 0 111 11 10 111 111 11 111 0 0 0 0 0 01 0 0 0 0 0 0 0 0 0 0 0 111 11 10 111 111 11 111 e2 0 0 0 0 0 01 0 0 0 0 0 0 0 0 0 0 0 111 11 10 111 111 11 111 0 0 0 0 0 01 0 0 0 0 0 0 0 0 0 0 111 11 10 111 111 11 11 0 0 0 0 0 01 0 0 0 0 0 0 0 0 0 111 11 10 111 111 11 1 0 0 0 01 0 0 0 0 0 0 0 0 1 11 10 111 111 11 11 1 11 10 111 111 11 0 0 0 0 0 01 0 0 0 0 0 0 0 0 11 1 11 10 111 111 1 0 0 0 0 0 01 0 0 0 0 0 0 0 11 1 11 10 111 111 0 0 0 0 0 01 0 0 0 0 0 0 11 1 11 10 111 111 0 0 0 0 0 01 0 0 0 0 0 0 11 1 11 10 111 11 0 0 0 0 0 01 0 0 0 0 0 11 1 11 10 111 1 0 0 0 0 0 01 0 0 0 0 11 1 11 10 111 0 0 0 0 0 01 0 0 0 11 1 11 10 111 0 0 0 0 0 01 0 0 0 11 1 11 10 11 0 0 0 0 0 01 0 0 11 1 11 10 1 0 0 0 0 0 01 0 11 1 11 10 0 0 0 0 0 01 11 1 11 10 0 0 0 0 0 01 11 1 11 1 0 0 0 0 0 01 11 1 11 00 0 00 11 1 00 0 11Figure 3.2: Two examples0 0 11 1 0 0 0 0 of eigenspaces extracted by PCA. The first principle eigenvector 11 1 00 01 11 1 00 0 11 00 11 00 1 0e1 is marked with a bold line and the second e2 (e1 ⊥ e2 ) with a dashed line. Example(a) demonstrates why PCA can be used for dimensionality reduction before classification.The projection on the e1 direction captures all information needed for classification. Incontrast, example (b) indicates a PCA drawback. Classification after projecting data toe1 direction is impossible.drawback of the PCA technique, namely a high sensitivity to the scaling that changesthe ordering of the eigenvectors. Scaling affects the reduced low-dimensional eigenspace,
    • Chapter 3: Reconstruction constraints 37extracted by PCA and being optimal for reconstruction it may be inappropriate for recog-nition. When the data is whitened, PCA is not clear at all, since all orthogonal systemsare equivalent from a PCA viewpoint.Autoencoder network The autoencoder networks have been successfully used notonly for compression (Mougeot et al., 1991; Cottrell et al., 1987), but for classification aswell (Elman and Zipser, 1988; Japkowicz et al., 1995; Schwenk and Milgram, 1995). Inthese works, a classification process is considered to consist of two phases. In the firstphase several autoencoders are trained. Each autoencoder is trained separately on thesamples of the corresponding class. The second phase is heuristic and is based on the ideathat the reconstruction error is, in general, much lower for examples of the learned classthan for the other ones. In (Japkowicz et al., 1995) classification is constrained to a two-class discriminationtask that is replaced by a dual task of familiarity with a concept. In the first phase, thesingle autoencoder is trained on the conceptual examples solely. In the second phase, theconceptual examples or two classes examples are used to estimate the decision thresholdfor a reconstruction error (the sum of squared errors) between input and output. Ifthe reconstruction error is smaller than the decision threshold, the instance is classifiedas conceptual, if larger it is classified as counter-conceptual. Similarly, in (Elman andZipser, 1988) the autoencoder is trained on segmented sounds that allows to segment acontinuous speech on the base of the mean squared error. In (Schwenk and Milgram, 1995) the basic idea is to use one autoencoder for eachclass and to train it only with examples of the corresponding class. In contrast with theusual autoencoder, a tangent distance is used instead of the squared reconstruction error.This tangent distance allows to incorporate a high-level knowledge about typical inputtransformation. Classification is done using the reconstruction errors of the autoencodersas discriminant functions.“Wake-sleep” network Another example of classification based on the reconstructionhas been proposed via the “wake-sleep” network (Hinton et al., 1995). Similarly to au-toencoders, each “wake-sleep” network is trained separately on different examples of thesame digit. Classification is done by observing which of the networks provides the mosteconomical description of the data.
    • Chapter 3: Reconstruction constraints 383.1.5 Other applications of reconstructionReconstruction via a modified autoencoder has been used for input reconstruction relia-bility estimation (IRRE) for autonomous car navigation (Pomerleau, 1993). In IRRE aconnectionist network is trained simultaneously to produce the correct steering responsefor a car navigation and to reconstruct the input image in the mean squared error sense.After learning, the reliability measure which is a correlation between the input and itsreconstructed image is evaluated. This reliability measure may be used to control vehiclespeed and its location in the a priori known confusing situations. Another applicationof IRRE is by integrating the outputs of multiple networks trained for different drivingsituations, i.e. the network that has the best reliability has to be used for a navigationtask. Another related recurrent network has been used for autonomous vehicle navigation(Baluja and Pomerleau, 1995). Baluja et al. use prediction of the next future input imageas a related task to the navigation task, i.e. the MLP network is learned to predict aninput image and to produce a right steering response simultaneously. Computationally,the hidden weights are updated based on the navigation task only, but from the obtainedhidden activities the network is trained to predict. Recursion has a place by propagatingthe predicted image back to the input layer for refining the next input image via noiseand unpredicted object elimination. A similar to IRRE connectionist network has been proposed as a hippocampal model(Gluck and Myers, 1993). This model assumes that the hippocampal region developsstimulus internal representation that enhances the discrimination of predictive cues whilecompressing the representation of redundant cues.3.2 Imposing reconstruction constraints3.2.1 Reconstruction as a bias imposing mechanismWe have shown above that the reconstruction task is related to the classification taskand two main approaches to classification via reconstruction take place. The first ap-proach offers the use of a common hidden representation obtained for all data as a pre-processing step for the following learning (Kirby and Sirovich, 1990; Murase and Nayar,1993; Moghaddam and Pentland., 1994). In the second approach (Japkowicz et al., 1995;Schwenk and Milgram, 1995; Hinton et al., 1995), reconstruction (generative) networksare used to extract the underlying structure of the data drawn from the same class.The assumption is that an example drawn from another class does not share the already
    • Chapter 3: Reconstruction constraints 39learned structure and produces a high description length. Thus, the description lengthmay be used as a discriminant function. Though these approaches have been relatively successful, there are cases when they arenot appropriate. For example, when the samples belonging to the same class have multi-modal distribution, or the classes are very similar, the second approach is not obvious.As we have shown in Section 3.1.4, PCA is very sensitive to data scaling. This consideration favors the view that each perceptual task needs data preprocessingthat can not be obtained based only on the other related task. Contrary to the consideredabove approaches, we propose to use reconstruction realized via a modified autoencoder asa bias-imposing mechanism in the feed-forward networks for improving the classificationtask. An intuitive way to conceive the idea of imposing reconstruction as a proper biasconstraint for classification is via the multi-task learning (MTL) approach (2.3.3). Ashas been shown above both recognition and reconstruction are related but different tasksof visual processing. In some cases, they were also replaced by one another. Secondly,it has been experimentally shown (Elman and Zipser, 1988; Cottrell et al., 1987), thatreconstruction via an autoencoder extracts a valuable internal representation. Thus, it isreasonable that hidden representation that relies on recognition and reconstruction taskscan improve the generalization performance of classification. This assumes that suchhidden representation has to capture some prominent (recognition) features of the data,while keeping most important information needed for reconstruction. As an illustration, let us assume that we want to classify between two individuals andsuppose that one of them has some prominent features in the training images (glasses, hairstyle, moustache, beard and so on), then it seems plausible that recognition will exhibita tendency to process these corresponding areas of the face and all the other informationwill be redundant for the recognition goal. However, these features may be absent orappear rarely in new images of this person, thus failure in the testing phase is likely. Incontrast, the addition of the reconstruction task during training of the system, forces thesystem to extract other features which may not be so useful for recognition of the originaltraining images, but may be of use with the novel test set. This motivates our suggestionto add reconstruction constraints during learning of the classification task. Similar approach has been proposed in (Gluck and Myers, 1993) to model a hip-pocampus function. It is assumed that one of the roles of the hippocampus is to extract acommon recognition/reconstruction internal representation of the input stimulus. Thoughconceptually our work is close to this model we have remarkable differences that are elu-cidated later on. Below, we present a hybrid classification/reconstruction network.
    • Chapter 3: Reconstruction constraints 403.2.2 Hybrid classification/reconstruction networkFigure 3.3 presents the architecture of the combined classification/reconstruction network.This network attempts to improve the low dimensional representation by minimizingconcurrently the mean squared error (MSE) of reconstruction and classification outputs.In other words, it attempts to improve the quality of the hidden layer representation byimposing a feature selection useful for both tasks, classification and reconstruction. Thehidden layer should have a smaller number of units compared with the input, so as toachieve a bottleneck compression and to allow for generalization. The combined learning Combined recognition/reconstruction network Reconstruction Input Hidden layer ClassificationFigure 3.3: A single hidden layer drives the classification layer and the reconstructionlayer.rule for the hidden layer units is a composition of the errors backpropagated from bothreconstruction and recognition layers. The relative influence of each of the output layersis determined by a constant λ which represents a tradeoff between reconstruction andclassification confidence. Below, we present a rigorous mathematical explanation of the hybrid network in theMDL framework.3.2.3 Hybrid network and MDLIt is easy to see that the proposed network is a modified autoencoder network. The mod-ified autoencoder shares a common hidden representation with the supervised (classifica-tion) network. It finds the compact hidden representation that is good for reconstructionin addition to a task at hand (Figure 3.4). In contrast to the autoencoder (Section 3.1.2)
    • Chapter 3: Reconstruction constraints 41 Hybrid network with reconstruction and EPP constraints Reconstruction Output 2 hidden-to-top 1 W W weights Bias constraints Hidden representation - A w - hidden weights Input - XFigure 3.4: The hidden layer drives the reconstruction and classification output. Inaddition, the search of another statistical structure in the data is made.and supervised feed-forward network (Section 2.7), the hybrid network is associated witha different communication game, in which the sender uses a compact internal representa-tion to communicate both the observed data and the corresponding desired output (forexample, class labels of the images). Since this internal representation has to encodeefficiently both the observed data and corresponding output, a cost for communicatingthe input data X has to be involved in the description length (2.7.24), yielding: L(M, D, X ) = − log p(D, X |w, W1 , W2 , M ) − log p(w, W1 , W2 |M ) + const.Assuming that given the input and the net weights, conditional probabilities of the recon-struction and supervised outputs are independent and Gaussian, we get similar to (2.7.29)the expression for the description length: 1 L(M, D, X ) = (λ1 ED + λ2 EX ) − log p(w, W1 , W2 |M ) + 2 r1 d log C(λ1 ) + r2 n log C(λ2 ) + const. (3.2.2)In expression for the description length (3.2.2), λ1 and λ2 are inversely proportional tothe variances of the specific task and reconstruction outputs respectively; ED and EX aresums of the squared errors of the supervised task and reconstruction outputs, respectively;r1 , r2 are numbers of training samples for reconstruction and specific tasks. Assuming the same a-priori probability for the hidden weights as in (2.7.30), the entire
    • Chapter 3: Reconstruction constraints 42description length may be simplified to: 1 L(M, D, X ) = (λ1 ED + λ2 EX + µE[H(w, x)]) + 2 (1) r1 d log C(λ1 ) + r2 n log C(λ2 ) + log CH (µ) + const . (3.2.3) (2) In general, the numbers of training samples for reconstruction (r1 ) and specific (r2 )tasks may be different, which seems to be a common situation in a real-world learning.In the limit, when we do not have enough information provided by supervised learning,internal representation is constructed based on the unsupervised learning only. Since, in our consideration, parameters λ1 , λ2 , µ are assumed to be fixed, the secondpart of the description length (3.2.3) is a constant and the description length may berewritten as: L(M, D, X ) = 1 (λ1 ED + λ2 EX + µE[H(w, x)]) + Const 2 (3.2.4)Therefore, when one is interested in both tasks, two scaled sum-square errors ED and EXpresent the error cost and the third term µE[H(w, x)] is the model-cost or representation-cost. This interpretation of the hybrid network in the MDL framework is not single.Indeed, an interpretation depends on a way to look at the hybrid network. When one is mainly interested in the reconstruction via bottleneck hybrid structure,the task may be formulated as a compression problem. This compression has benefitscompared to a conventional autoencoder, since it admits not only a good reconstructionof the data, but a successful handling on the specific task, such as classification, forexample. In this statement, the reconstruction error EX is recognized as the error costand the scaled classification error and third term as the model-cost. The third and the last interpretation is produced when one is mainly interested inthe specific task (for example, classification or control tasks). In this case, the specificerror ED is recognized as the error cost and the scaled reconstruction error and third termas the model-cost. This interpretation gives the rigorous mathematical way for imposingreconstruction and other unsupervised types of constraints in the supervised network. Below, based on this last interpretation, we explain why the hybrid network may bebetter than the conventional classification feed-forward network. Let us consider twodifferent principal melodies. Suppose that the first is embellished with specific tones,however, the second is not arranged at all. Hearing these two melodies, arranged and not,many times, one can decide that these specific tones are enough for recognizing, which
    • Chapter 3: Reconstruction constraints 43one of the melodies is played. However, the next time the first melody may be played bya non skilled pianist, that skips all the beautiful ornaments. Obviously, in this case, thefirst melody will never be recognized, based on the presence of the ornaments only. This example demonstrates that a bottleneck network for classification, attempting tominimize the description length has a tendency to throw away salient information fromthe data. The internal representation extracted, based on the classification task alone,may be too poor. Reconstruction helps to process information as a whole, it does notconcentrate on the particular details, balances the relationship between the whole and itsparts, resulting in a better prediction on the supervised specific task.Bayesian interpretation for the hybrid NNWe have shown that the MDL approach naturally explains and interprets the proposed hy-brid classification/reconstruction network. It also states that the most probable networkweights have to minimize the following part of the description length (3.2.4): R(w, W1 , W2 ) = λ1 ED (w, W1 ) + λ2 EX (w, W2 ) + λ3 H(w). (3.2.5)We recall now that the MDL principle is tightly related to the Bayesian approach, whereparameters λ1 , λ2 , µ are recognized as hyper-parameters. When the hyper-parametersare unknown, the Bayesian correct treatment (Bishop, 1995a) is to integrate the hyper-parameters out of any predictions P: p(P|D, X ) = p(P|D, X , λ)p(λ|D, X )dλ, (3.2.6)where λ = (λ1 , λ2 , µ) is a vector of hyper-parameters and p(λ|D, X ) is the evidencefor the hyper-parameters. This integration is similar to generating an ensemble fromthe networks which depend on the hyper-parameters, where instead of evaluation of thehyper-parameter evidences (that is impossible analytically), we integrate predictions inthe vicinity of the most likely hyper-parameters assuming equal evidences. Thus, contrary to (Gluck and Myers, 1993; Pomerleau, 1993), we do not considersome fixed manually adjusted parameters, but a class of the reconstruction-classificationnetworks depended on the “regularization” parameter, with the subsequent combinationof the networks to ensembles.3.2.4 Hybrid network as a generative probabilistic modelBoth “recognition” and “generative” phases can be identified in the proposed hybridmodel. The “recognition” phase infers an internal/hidden representation of the input data.
    • Chapter 3: Reconstruction constraints 44The “generative” phase reconstructs the input from the inferred compact representationin the reconstruction output sublayer and, in addition, predicts the specific task outputin the corresponding sublayer. From a Bayesian viewpoint, learning in the hybrid network is equivalent to maximiza-tion of the joint probability of the input and specific task output, given the observationand specific constraints on the internal representation. According to Bayesian theory, thebest classification is based on the conditional probability of the image classes given theinput (i.e. the conditional probabilities of the image classes are the best discriminantfunctions, that lead to the minimal classification error). The output of the recognitionlayer of the hybrid network estimates this conditional probability and the reconstructionsublayer regenerates the input data, implicitly estimating the probability of the inputdata. The proposed architecture differs from a probability network that has a generativemodel (reconstruction) and a recognition model in a manner similar to the binary wake/sleeparchitecture (Hinton et al., 1995), or the Rectified Gaussian Belief Network (Hinton andGhahramani, 1997). First, it is not a full forward/backward model, namely there are notwo hidden unit representations, one for the top-down and one for the bottom-up, butinstead a single hidden representation is used for both (Figure 3.3). Second, its learninggoal is to minimize the classification error (via the mean squared error) as well as tominimize the reconstruction error, as opposed to the goal of constructing a probabilisticmodel of internal representations. The two goals may coincide under a continuous hiddenunit network, but are certainly different for a binary network.3.2.5 Hybrid Neural Network architectureThe detailed architecture of the network is presented in Figure 3.5. This hybrid networkis a modification of the well-known feed-forward network. It is supplied by images in theinput layer which are propagated via a hidden layer to the output layer. The output layerconsists of two sub-layers, one sub-layer reconstructs the image, the second one servesfor classification. The number of units in the output reconstruction sub-layer and theinput layer are the number of pixels in the image. The hidden layer has a smaller numberof units, because we are looking for aggressive compression techniques to overcome the“curse of dimensionality”. The output classification layer has a number of units equal tothe number of image classes. Each image is propagated to the hidden layer in the form: N hj = wji xi + wj0 (3.2.7) i=1 yj = σ(hj ), j = 1, . . . , m, (3.2.8)
    • Chapter 3: Reconstruction constraints 45 Detailed architecture of the recognition/reconstruction network Reconstructed image. Reconstruction sublayer Recognition sublayer. (s=1,..K) Xi Ps 2 Wij 1 Wsj Yj (j=1,...,m) Hj Wji Xi (i=1,...,n) Input image.Figure 3.5: Feed-forward Neural Network with recognition and reconstruction outputsub-layerswhere m N is the number of hidden units, N is the number of pixels in the image, σis the sigmoid activation function: 1 σ(x) = . (3.2.9) 1 + exp(−x)Image reconstruction, based on the hidden layer representation, is given by: m 2 2 xi = ˆ Wij yj + Wi0 , i = 1, . . . , N, (3.2.10) j=1and the output of the recognition layer unit is calculated according to the formula: m 1 1 ps = σ( Wsj yj + Ws0 ), s = 1, . . . , K, (3.2.11) j=1
    • Chapter 3: Reconstruction constraints 46where K is the number of individuals (number of classes). The classification is madeaccording to the maximal response of the recognition sub-layer (ps is interpreted as theprobability of the sample to belong to a certain class -s).3.2.6 Network learning ruleLet us consider the error back-propagation learning rule with a goal of minimizing thecost function, which is a weighted sum of scaled recognition and reconstruction errorswith coefficients λ1 and λ2 , respectively1 : E(w, W1 , W2 ) = λ1 E 1 (w, W1 )/K + λ2 E 2 (w, W2 )/N. (3.2.12)As has been shown in the Section 3.2.3, the coefficients λ1 and λ2 are inversely propor-tional to the noise variances in the reconstruction and recognition channels, respectively.Therefore, the larger the noise is in the channel, the less is the weight of the error-costcorresponding to this channel. Recognition E 1 and reconstruction E 2 errors, sum squaredover all samples are given by: M K E1 = (pµ − tµ )2 . s s (3.2.13) µ=1 s=1 M N 2 E = (ˆµ − t(xµ ))2 xi i (3.2.14) µ=1 i=1In this expression, t(xµ ) determines the target of the µ-sample in the reconstruction iunit-i. The most reasonable choice for t(xµ ), not demanding any a-priory knowledge, is it(xµ ) = xµ . Correspondingly, tµ is a target for recognition given by: i i s 1 if s coincides with class of µ-sample tµ = s 0 otherwise The weights between output-to-hidden and hidden-to-input layers update accordingto the gradient descent rule : ∆W1 = −η λ1 W1 E 1 (w, W1 ) (3.2.15) ∆W2 = −η λ2 W2 E 2 (w, W2 ) (3.2.16) ∆w = −η (λ1 w E 1 + λ2 w E 2) (3.2.17) 1 For convenience, scaling of errors E 1 and E 2 by the number of pixels in the image N , and the numberof image classes K respectively is carried on. This scaling serves to balance the values of the recognitionand reconstruction errors in Eq. 3.2.4.
    • Chapter 3: Reconstruction constraints 47Specifically, the weights between output reconstruction-to-hidden layers are given by: M 2 µ,2 µ ∆Wij = ηλ2 δi yj (3.2.18) µ=1 δi = ∆µ,2 ≡ (xµ − xµ )/N µ,2 i i ˆi (3.2.19) i = 1, . . . , N , j = 1, . . . , m, µwhere yj (3.2.8) is the output of the hidden unit-j in the feed-forward propagation of the µ,2input image-µ, and δi is the image reconstruction error, scaled by the number of pixelsin the image. Similarly, the weights between output recognition-to-hidden layers changeby: M 1 µ,1 µ ∆Wsj = ηλ1 δs yj (3.2.20) µ=1 m µ,1 1 µ δs = σ ( Wsj yj + Ws0 )∆µ,1 1 s (3.2.21) j=1 s = 1, . . . , K , j = 1, . . . , mwhere ∆µ,1 -recognition (regression) error scaled by the number of image classes: s ∆µ,1 = (tµ − pµ )/K. s s s (3.2.22) We call δ the output error of the layer and ∆ the input error to the layer in thebackward propagation. According to the generalized delta-rule (Hertz et al., 1991), thechange of deeper embedded weights between hidden-to-input layers has the form: M ∆wji = η δj xµ µ i (3.2.23) µ=1 µand the output error of the hidden unit-j δj in the backward propagation of the error isgiven by: δj = σ (hµ )∆µ . µ j j (3.2.24)Input error to the hidden unit-j ∆µ has the form: j ∆µ = λ1 ∆µ,1 + λ2 ∆µ,2 j j j (3.2.25) N ∆µ,1 = j 1 µ,1 Wij δi (3.2.26) i=1 K ∆µ,2 = j 2 µ,2 Wsj δs . (3.2.27) s=1
    • Chapter 3: Reconstruction constraints 48 µFrom (3.2.25–3.2.27) it is easy to see that the output error δj may be written as thesum of the errors back propagated concurrently from the reconstruction and recognitionsub-layers: µ µ,1 µ,2 δj = λ1 δj + λ2 δj (3.2.28) δj = σ (hµ )∆µ,1 µ,1 j j (3.2.29) δj = σ (hµ )∆µ,2 . µ,2 j j (3.2.30) In general, the input/output errors to any layer of the network are a weighted sumof the input/output errors back propagated from the lateral sub-layers (a chain rule ofthe derivatives). Thus, in the error back-propagation mode, hybrid network with lateralsub-layers emerges as a linear superposition of the conventional (classical) subnetworks.3.2.7 Hybrid learning rule.We follow the gradient descent algorithm as the errors are back propagated from an inputlayer to a hidden layer with a properly scaled cost function (3.2.12): E(w, W1 , W2 ) = (1 − λ)E 1 (w, W1 )/K + λE 2 (w, W2 )/N, (3.2.31)where λ ∈ [0, 1] (λ = λ2 /(λ1 + λ2 )) is a regularization parameter, which represents atradeoff between reconstruction and classification confidences. According to the gradient descent method, updating of the weight vector in eachiteration has to be done in the direction that has a negative projection on the gradientdirection. This permits us to rescale a learning rule (3.2.15-3.2.16): ∆W1 = −η W1 E 1 (w, W1 ) ∆W2 = −η W2 E 2 (w, W2 ) ∆w = −η ((1 − λ) w E1 + λ w E 2) (3.2.32) We emphasize that the parameter λ in our implementation, affects only the weights wbetween input and hidden layers, i.e. on the hidden layer representation. Our rule (3.2.32)may be treated as the hidden layer belief in the performance of the two upper channels,transferring backward information from reconstruction and recognition sub-layers. Thus,we take the errors of the reconstruction layer with the weight λ, and the errors of therecognition layer with the weight 1 − λ. It can be seen, that for λ = 0 the hiddenrepresentation is built based only on the recognition task, and reconstruction is learnedfrom the hidden layer. This marginal case corresponds to the Baluja consideration (Balujaand Pomerleau, 1995). In contrast, when λ = 1, the hidden representation is based on the
    • Chapter 3: Reconstruction constraints 49reconstruction task solely; and we attempt to solve the recognition task in the reducedspace. We see that this marginal case is equivalent to a first approach to classify viareconstruction (Kirby and Sirovich, 1990; Turk and Pentland, 1991; Murase and Nayar,1993). This network and its hybrid rule may be interpreted as the parallel concurrent workof two separate feed-forward networks for recognition and reconstruction. The hybrid nethidden weight updating is a linear combination of the gradient directions of both networksin the common hidden weight space. For small λ our method is a kind of gradient descentmethod that prevents zig-zags (peculiar to the gradient steepest descent method (Ripley,1996)) in the search of the optimal weights minimizing the recognition regression error.
    • Chapter 4Imposing bias via unsupervisedlearning constraints4.1 IntroductionInformation theory provides some explanation to sensory processing (Rieke et al., 1996).According to these principles, neural cell responses are developed by optimizing criterionsbased on the information theory. The first proposed information principles are redundancyreduction (Barlow, 1961) and “infomax” (Linsker, 1988), that are similar and lead to afactorial code formation under some conditions (Nadal and Parga, 1994). Recently, withthe parallel development of independent component analysis (ICA) (Comon, 1994) inthe signal processing, new efficient algorithms for the factorial code formation have beenproposed. Of particular interest are algorithms via feed-forward networks with no hiddenlayer (Bell and Sejnowski, 1995; Yang and Amari, 1997). In this chapter, we propose to use information theoretical measures as constraints forthe classification task. We introduce a hybrid neural network with a hidden representa-tion that is arranged mainly for the classification task and, in addition, has some usefulproperties, such as the independence of hidden neurons or maximum information transferin the hidden layer, etc. The chapter is organized as follows. In the first section, the main information principlesand their relation to sensory processing are discussed. The second section presents themathematical background and algorithms for ICA and other related information prin-ciples. In the third section, a hybrid neural network with unsupervised constraints isintroduced and some algorithmical details are presented. 50
    • Chapter 4: Unsupervised learning constraints 514.2 Information principles for sensory processingMammals process incomplete and noisy sensory information in an apparently effortlessway. This is possible since sensory inputs: images, sounds, etc., have very specificstatistical properties that are efficiently encoded by the biological nervous systems. Thesensory inputs appear usually smooth over large spatial and temporal regions that leadto redundancy in the sensory input. The redundancy emerges as a statistical regularity,which means that many pieces of a signal are a-priori predictable from other pieces andhence by clever recoding it is possible to get more economical representation of the data. In the past, the principle of redundancy reduction (Barlow, 1961) was suggested as acoding strategy in neurons. According to this principle each neuron should encode featuresthat are as statistically independent as possible from other neurons over a natural ensembleof inputs. The ultimate obtained representation is called the factorial code (Redlich, 1993).In the factorial code, the multivariate probability density function (pdf) is factorized asa product of marginal pdfs. This property provides an efficient way of storing statistical 1knowledge about the input (Barlow, 1989). One of the earliest attempts to construct the factorial representation via neural net-works was proposed by Atick (1992). The underlying computational learning rule is basedon the minimization of the sum of the entropies of the hidden units under constraint topreserve the input entropy (the total information about the signal). A type of gradientdescent algorithm in the assumption of a Gaussian input signal and linear output, re-sults in a Hebbian-like learning rule and a decorrelated hidden representation. The majorlimitation of Hebbian-like rules is dependence on the linear, pairwise correlations amongimage pixels (second order statistics). Thus, they are not sensitive to phase changes inthe image responsible for oriented localized structures, such as lines, edges and corners(Field, 1994). Motivated by the principle of redundancy reduction Field (1994) contrasts two differentcoding approaches. Both approaches take advantage of the input redundancy, but in adifferent manner. The first one, compact coding, is based on the mean-squared error anduses only the second order statistics of the input. The main goal of this coding is toreduce dimensionality of the input in the directions with a low input variance. PCAand linear auto-associator networks, considered in Chapter 3, are examples of this codingscheme. An alternative sparse distributed coding does not necessarily imply the reductionof dimensionality. In contrast, the dimensionality may be enlarged. A sparse distributed 1 For an image description the probability of each possible set of pixel values has to be known. Forinstance, an image having N pixels with Q intensity quantization levels requires the storing of QN possibleprobabilities. If the code is factorial the number of the required probabilities reduces to N Q.
    • Chapter 4: Unsupervised learning constraints 52coding approach encourages representations, where only a small, adaptive to input, subsetof hidden units is simultaneously active. Although, there is not a general tool to form the sparse code, it has some typicalfeatures. The sparse code is characterized by the extremely peaked distribution of thehidden unit activities which provides both high probability of a neuron to be silent oractive according to its relevance to the input pattern representation. A way to constructsparse coding based on this feature has been proposed in (Olshausen and Field, 1996) byminimizing the cost functional consisting of a mean-squared error and a penalty term forneuron activities. Peaked distributions are characterized by high kurtosis or low entropies (Oja, 1995b),thus, maximization of kurtosis or entropy minimization can be used for sparse codingformation. At the same time, via minimization of the sum of the entropies sparse codingis related to a factorial coding. It is also known that under a fixed variance the Gaussiandistribution has the largest entropy (Cover and Thomas, 1991). Thus, hidden unit entropyminimization is tightly related to exploratory projection pursuit (EPP), which tries tofind a structure in the projected data, seeking directions that are as far from Gaussian aspossible (Friedman, 1987). Therefore, a deviation from the Gaussian distribution servesas a good measure for hidden unit independence and can be used as a strategy for sparsecoding construction. Recently, an interest in EPP has been revived and formulation of the new unsuper-vised rules based on the information theory has been stimulated with the development ofindependent component analysis (ICA). In the next section, ICA is formulated and somealgorithms producing factorial codes are presented.4.3 Mathematical backgroundICA has been developed as a tool for blind source separation. The problem is to recoverindependent sources from sensory observations which are unknown linear mixtures of theunobserved independent source signals. Let us consider m unknown mutually independentsources si (t), i = 1, . . . , m with no more than one being normally distributed. In general,t is a sampling variable, that may be a time variable for signals or a two dimensionalspatial variable for images, or an index of the pattern in a data-set. The sources aremixed together linearly by an unknown non-singular matrix A ∈ Rn×m : x(t) = As(t), s(t) = [s1 (t), . . . , sm (t)] (4.3.1)It is assumed that in (4.3.1) the number of sensors xi (t), i = 1, . . . n is greater or equalto the number of sources (n ≥ m). The task is to recover the original signals via a linear
    • Chapter 4: Unsupervised learning constraints 53transform defined by a matrix W ∈ Rm×n : u(t) = Wx(t), u(t) = [u1 (t), . . . , um (t)] (4.3.2)Since recovered signals may be permuted and scaled versions of the sources, the de-mixingmatrix W has to be a solution of the following linear equation: ΛP = WA,where Λ is a non-singular diagonal matrix and P is a permutation matrix.4.3.1 Entropy maximization (ME)One of the first algorithms extracting the independent components via a neural networkhas been proposed by Bell et al. (1995). Assuming that the number of sources is equalto the number of sensors, a fully connected n → n feed-forward network consisting froman input and nonlinear output layers, having the same number of units as the number ofsources, has been considered (Figure 4.1). The network has been trained to maximize a Feed-forward network for independent component extraction Output y: y i =g i (u i ) yi ui u=Wx (u-recovered sources) Input - x Figure 4.1: A one layer n → n feed-forward network.joint entropy H(y) of the nonlinear output y: u = Wx + w0 , y = g(u), y ∈ Rn , u ∈ Rn , w0 ∈ Rn (4.3.3) H(y) = − p(y) log p(y)dyIn the case of the output additive noise, the entropy maximization (ME) is equivalentto maximization of the mutual information between input and output (Nadal and Parga,
    • Chapter 4: Unsupervised learning constraints 541994). As has been shown earlier (Linsker, 1988), the principle of the mutual informa-tion maximization called “infomax” in the case of a linear neural network leads to aHebbian like learning rule, that is sensitive to the second order statistics only, therefore,nonlinearity in the output layer is essential. The joint entropy of the output can be represented as: H(y) = H(yi ) − I(y), (4.3.4) iwhere H(yi ) = − p(y i ) log p(y i )dy i are marginal entropies of the outputs and I(y) istheir mutual information. The mutual information (MI) of the output y is a Kullback-Leibler measure between output distribution p(y) and a product of marginal distributions i p(yi ): p(y) I(y) = p(y) log dy (4.3.5) i p(yi )Due to a ∩-convexity of the log function, the Kullback-Leibler measure is nonnegative andattains its minimum zero value if and only if outputs yi are independent almost every-where. Maximization of the joint entropy consists of maximizing the marginal entropiesand minimizing the mutual information. Since the nonlinear functions bound the outputs,the marginal entropies are maximum for a uniform distribution of yi . The mutual infor-mation I(y) is invariant under an invertible component-wise transform (I(y) = I(u)) andachieves its minimum equal zero when the presynaptic outputs u (4.3.3) are independent.Thus, if the nonlinear functions gi have the form of the cumulative density function (cdfs)of the true source distribution, then the matrix W recovers independent sources as thepresynaptic output u (4.3.3), and this is a single global maximum of the joint entropyH(y), which is a convex ∩ function. As has been rigorously proven (Yang and Amari, 1997), the ME approach leads tothe independent components only if the nonlinear activation functions gi in the outputlayer coincide with the cumulative density functions (cdfs) of the sources. For zero meanmixtures and functions gi not equal to the (cdfs) of the sources, the ME algorithm doesnot converge to the ICA solution W = ΛPA−1 . However, if the initial matrix is the right −1ICA solution W0 = ΛPA , the algorithm does not update the de-mixing matrix W inthe directions of increasing the cross-talking. This fact partially explains the ME success,even when cdfs are not known exactly. In applications considered by Bell and Sejnowski(1995), nonlinear activation functions have been chosen ad hoc as logistic sigmoidal, thathas a highly peaked derivative with long tails. Since sound signals are super-Gaussians2this type of nonlinearity appears to be appropriate for “infomax” principle. 2 Super-Gaussian signals have pdf with large tail areas and a sharp peak. In contrast, sub-Gaussiansignals have pdf with small tail areas and a flat peak (see also Appendix A to Chapter 4.)
    • Chapter 4: Unsupervised learning constraints 55 The de-mixing matrix W is found as synaptic weights of the network iteratively usingthe stochastic gradient ascent method applied to the joint entropy H(y): ∆W = η([Wt ]−1 + (1 − 2y)xt ) 1 ∈ Rn , ∆w0 = η(1 − 2y) Amari et al. (1997) have suggested a modification of this rule that utilizes the nat-ural gradient and does not require the inversion of the weight matrix. It proceeds bymultiplying the absolute gradient by Wt W, producing3 : ∆W = η(I + (1 − 2y)ut )W (4.3.6)4.3.2 Minimization of the output mutual information (MMI)Another way to derive independent outputs for the blind separation problem has beenpresented in (Amari et al., 1996). An algorithm minimizes the mutual information (MI)of the linear outputs, Iu (W): n Iu (W) = −H(u) + H(ui ), (4.3.7) i=1 u = Wx, (4.3.8)that attains its minimum if and only if the outputs ui are independent about everywhere.In order to approximate marginal entropies H(ui ), truncated Gram-Charlier expansion(Stuart and Ord, 1994) of the marginal pdfs p(ui ) has been used and a mild assumptionabout the original source statistics has been done. It has been assumed that the originalsources have zero mean and their variances are normalized to 1. A stochastic gradientdescent applied to the approximated expression of the mutual information Iu (W) leadsto the following equation for the network weight dynamics: ∆W = η([Wt ]−1 − Φ(u)xt ) (4.3.9)where Φ(u) = f (k3 , k4 ) ◦ u2 + g(k3 , k4 ) ◦ u3 and the following notations have a place: f ◦ y = [f1 y1 , . . . , fn yn ]t , uk = u ◦ uk−1 f (k3 , k4 ) = [f (k3 , k4 ), . . . , f (k3 , k4 )]t , g(k3 , k4 ) = [g(k3 , k4 ), . . . , g(k3 , k4 )]t 1 1 n n 1 1 n n k3 = mi = E[u3 ], k4 = E[u4 ] − 3(E[u2 ])2 i 3 i i i i 1 9 1 3 3 f (a, b) = − a + ab, g(a, b) = − b + a2 + b2 2 4 6 2 4 3 w0 is assumed to be zero.
    • Chapter 4: Unsupervised learning constraints 56The natural gradient descent for MMI leads to the following algorithm: ∆W = η(t)[I − Φ(u)ut ]W (4.3.10) As has been pointed out in (Yang and Amari, 1997), both ME and MMI algorithmshave the same typical form (4.3.10). In ME Φ depends on the nonlinear activationfunctions gi and is given by: g1 (u1 ) g (un ) t Φ(u) = −( ,..., n ) (4.3.11) g1 (u1 ) gn (un )Since gi should coincide with the cdfs of the unknown original signals, Φ(u) have tobe chosen properly. In MMI, functions Φ depend on the cumulants of the third and i ifourth orders k3 , k4 of the linear scalar output ui . These cumulants may be replaced byinstantaneous values or be estimated. Another possibility is to use a-priori knowledgeabout cumulants of the unknown original signals. Therefore, whereas a success of theME algorithm depends on the a-priori knowledge about data statistics, MMI is moreflexible. In (Yang and Amari, 1997) the following types of Φ(u) = (φ(u1 ), . . . , φ(un ))t havebeen used: (a) φ(u) = u3 (4.3.12) (b) φ(u) = tanh(u) (4.3.13) 3 15 14 29 29 (c) φ(u) = u11 + u9 − u7 − u5 + u3 (4.3.14) 4 4 3 4 4The (a-b) forms of Φ(u) correspond to the ME algorithm and assume pdfs (and equiva- 4lently g(u)) to be proportional to: (a) p(u) ∝ exp(−u4 /4) (4.3.15) (b) p(u) ∝ (cosh(u))−1 (4.3.16)Therefore in both cases, distributions are assumed to be symmetrical and sub-Gaussian.The form (4.3.14) of Φ(u) is the instantaneous form of MMI (k3 = u3 , k4 = u3 − 3) and i i i iit does not assume the shape of the source distributions. The ME and MMI learning rule (4.3.10) has been obtained in the assumption of asquare weight matrix, W. However, in some applications it may be interesting to separate u 4 Here we use the fact that − g (u) = φ(u), which leads to g (u) = exp(− g (u) 0 φ(u))du and at the sametime g (u) must coincide with pdfs of the original sources.
    • Chapter 4: Unsupervised learning constraints 57only a part of the sources. This may be done via multiplication of the right side of thelearning rule (4.3.9) by Wt ΛW, where the block matrix Λ ∈ Rn×n is given by : I 0 Λ= . (4.3.17) 0t 0sIn (4.3.17) I ∈ Rm×m is an identity square matrix, 0 ∈ Rm×(n−m) is a rectangular zeromatrix, 0s ∈ R(n−m)×(n−m) is a square zero matrix and m < n. The final learning rule for ˜a part of the weight matrix W, obtained by deleting the last (n − m) rows of the matrix ˜W will be the same one as (4.3.10): W ∈ Rm×n : ˜ ˜ ∆W = η(t)[I − Φ(u)ut ]W ˜ W ∈ Rm×n , Φ(u) ∈ Rm×1 , u ∈ Rm×1 , I ∈ Rm×mThe network architecture then implies dimensionality reduction since the number of out-put units is less than the number of input units. In addition, such a network extractsindependent components.4.3.3 Relation to Exploratory Projection Pursuit.MMI has been considered as the starting point for a large family of ICA contrast functionsproposed by Hyvarinen (Hyvarinen, 1997a). It has been noted that MI can be expressedusing negentropies J(u), J(ui ) 5 (Hyvarinen, 1997a; Girolami and Fyfe, 1996): 1 Cii Iu (W) = J(u) − J(ui ) + log i , (4.3.18) i 2 det(C)where C is a covariance matrix of u and Cii are its diagonal elements. Since the negentropyJ(u) is invariant for invertible linear transformations (J(u) = J(x), note that J(ui ) =J(xi ) holds only when nonlinear transformation: x → u, is componentwise with ui =f (xi )), MMI is roughly equivalent to finding directions in which negentropy is maximized.This equivalence is rigorous, when components ui are constrained to be uncorrelated (thelast term of 4.3.18 is zero). This means that the directions in which the data distributionis as non-Gaussian as possible are preferable. This is the point where EPP and ICA havecome into contact. The natural gradient ascent applied to the sum of the marginal negentropies leads tothe same learning rule (4.3.10) (Girolami and Fyfe, 1996; Lee et al., 1998). When the 5 Negentropy of the multivariate random variable u is a difference between entropies of the multivariateGaussian distribution with the same covariance matrix as u and entropy of the u: J(u) = H(uG )−H(u).It measures deviation of the distribution from Gaussian and is nonnegative. The valuable property ofnegentropy is invariance under invertible linear transforms.
    • Chapter 4: Unsupervised learning constraints 58nonlinearities Φ are taken to be: ui + tanh(ui ) for super-Gaussian source φi (ui ) = ui − tanh(ui ) for sub-Gaussian sourcethe learning rule may be written in the elegant form: ∆W = η(t)[I − K tanh(u)ut − uut ]W, (4.3.19)where K is a diagonal matrix with elements sign(kur(ui )) and kur(ui ) is kurtosis of thei-source. The advantage of EPP, however, is the possibility to find independent componentsrecursively one-by-one by maximization of the 1-D negentropy. For the same conditions,as in (Yang and Amari, 1997): E[ui ] = 0, E[u2 ] = 1, negentropy may be approximated iby: 1 1 (E(u3 ))2 + (k4 (ui ))2 J(ui ) ≈ i (4.3.20) 12 48When source distributions are assumed to be symmetrical, negentropy simplifies to J(ui ) ∝(k4 (ui ))2 = kuri and minimization of the output mutual information is approximately 2equivalent to maximization of the sum of the source kurtosises6 : 2 Fmax (W) = kuri (4.3.21)In other words, the directions in which signal distribution is highly peaked or extremelyflat, are considered as interesting. In (Hyvarinen, 1997b) a new family of approximated contrast ICA functions has beenproposed via the negentropy approximation: J(u) ∝ (E[G(u)] − E[G(ν)])2 , (4.3.22)where ν is a standardized Gaussian variable and the function G fulfills some orthogonalityproperty and is suitable to the assumed original source statistics and is reasonably simplefor computation. The simplest proposed choices for the function G is polynomial G = |u|α ,where α < 2 for super-Gaussian densities and α > 2 for sub-Gaussian densities. Thisapproach appears finally as a generalization of different projection pursuit indices (Blaiset al., 1998), where skewness and kurtosis are used explicitly to measure deviation fromthe Gaussian distribution. It is related also to the BCM neuron learning rule (Intratorand Cooper, 1992). 6 See Appendix A to Chapter 4 for cumulants and kurtosises definitions.
    • Chapter 4: Unsupervised learning constraints 594.3.4 BCMAn idea of BCM is to find a direction w which emphasizes data multi-modality by mini-mizing a specific loss function (a specific projection index): 1 1 F(w) = −µ( E[u3 ] − θ2 ) (4.3.23) 3 4 t 2 u = w x, θ = E[u ],In order to make this measure robust to outliers, a rectification nonlinear function isapplied in the linear output. Thus, in general, y = g(wt x). The gradient descent ruleyields the following learning rule: ∆w = µE[φ(y, θ)g (u)x] (4.3.24) where φ(y, θ) = y 2 − yθ, θ = E[y 2 ].4.3.5 Sum of entropies of the hidden unitsBeing motivated to obtain a hidden representation where each neuron contains as muchinformation as possible, we suggest to maximize the sum of the entropies of the outputunits: m F(W) = H(yi ). i=1The stochastic gradient descent method leads to the following equation for the weightdynamics (details are given in Appendix B to Chapter 4: g ∆W = η(f (u) + g )xt , (4.3.25)where f (u) is defined as φ(u) in (4.3.14). Since the nonlinear output functions bound theoutput values yi , the entropy is maximized, when yi is uniformly distributed, which leads dgito a relation pu (ui ) = dui . This means that the distribution of the presynaptic variablesui is controlled by the nonlinearities in the learning rule (4.3.25). For logistic sigmoidal activation functions, (4.3.25) simplifies to: ∆W = η(f (u) + (1 − 2y))xt (4.3.26)The same rule, but with the negative parameter η, can be used for a sparse code formation,as suggested in (Olshausen and Field, 1996; Atick, 1992)7 . 7 When the output is bounded c < y < d, due to a ∩-convexity of the log function, the entropy of the doutput is upper bounded: H(y) = c p(y) log p(y) dy ≤ log( p(y) )dy = log(d − c). Therefore, the entropy 1 p(y)maximization is properly defined mathematically. At the same time the lower estimate depends on the d ddistribution: −H(y) = c p(y) log p(y)dy ≤ log c (p(y))2 dy ≤ 2 log((d − c) max p(y)). It is clear thatin practice max p(y) is bounded and therefore, the problem of the sum of entropies minimization is alsoproperly defined mathematically.
    • Chapter 4: Unsupervised learning constraints 604.3.6 Nonlinear PCAAlthough the nonlinear PCA method has no apparent connection to the ME or MMI,it has been shown that it allows separation of the whitened linear mixtures of sources(Oja, 1995b; Oja, 1995a). In nonlinear PCA, the input signals are first prewhitened, i.e.the signals are represented as the projections on the eigenspace of the input covariancematrix and are properly scaled. As a result prewhitened signal x has a zero mean and aunit spherical covariance matrix. The learning rule is an approximate stochastic gradientdescent algorithm that minimizes the mean-squared reconstruction error: E = E[ x − Wt y 2 ], (4.3.27)where the weight matrix W and nonlinear output y are defined to be the same as in(4.3.3) and the bias is assumed to be zero w0 = 0. An approximate learning rule has the form: ∆W = ηy(xt − yt W) (4.3.28)For separation, odd twice differentiable nonlinear functions gi have to be properly takento satisfy some stability conditions depending on the data statistics. Particularly, it isshown (Oja, 1995a) that a sigmoidal nonlinear activation function as g = tanh(βu), β > 0is feasible for sub-Gaussian original signals and polynomial g = u3 for super-Gaussiandensities (in this analysis it was assumed that the sources are statistically identical andhave a symmetrical distribution). The MSE for whitened data and nonlinear activation functions in the form g(u) = u3or tanh(u) may be approximated as − kuri (Lee et al., 1998). Thus, minimization ofthe MSE leads to maximization of the sum of the kurtosises: Fmax (W) = kuri (4.3.29)The latter expression is equivalent to (4.3.21) for super-Gaussian original sources. Thisevaluation shows that in some cases the nonlinear PCA can also be viewed from information-theoretic principals, as a method to minimize approximately the mutual information ofthe output.4.3.7 Reconstruction issueLearning in the nonlinear PCA and nonlinear autoencoders is based on the reconstructionmean-squared error. Similarly to a linear case, nonlinear PCA and nonlinear autoencoderextract different weights. The nonlinear autoencoder with proper activation functions does
    • Chapter 4: Unsupervised learning constraints 61not necessary extract the independent components as the nonlinear PCA does in somecases. However, this consideration sheds light on the relation between the unsupervisedlearning based on the information theory and reconstruction. Reconstruction and ICA are related also via a generative model approach (MacKay,1996; Roweis and Ghahramani, 1997; Lee et al., 1998). ICA recovering independent com-ponents (hidden variables) and de-mixing weight matrix W is itself a recognition phaseof the reconstruction process, with a nonlinear generative model that differs from genera-tive models underlying PCA and linear autoencoder. Thus, although ICA (information-theoretic) constraints may be also considered as some type of “generalized” reconstructionconstraints with another underlying generative model, we keep the notion of reconstruc-tion constraints for an autoencoder network. ICA similar to PCA has been also used as a preprocessing step for face classification(Bartlett et al., 1998). As will be clear later, the hybrid classification/feature extractionscheme which is introduced in the next section corresponds to this type of preprocessing,when trade-off parameter λ = 1.4.4 Imposing unsupervised constraintsThe unsupervised learning rules we have used are based on different assumptions about thequality of the low dimensional representation (LDR). These rules are based on statisticsof order higher than two, and use low order moments of the distribution and a sigmoidalsquashing function for robustness against outliers. The learning rule for hidden weights modification for the constrained network (Fig-ure 2.2) is described by: ∆w = −η((1 − λ) w E 1 − λh(w, x)), (4.4.30)where the term h(w, x) corresponds to weight updating, that emerges via additionalunsupervised feature extraction. When h(w, x) is a gradient of some information measure 8H(w, x) the learning rule (4.4.30) corresponds to minimization of the penalized meansquared recognition error: F(w, W1 ) = (1 − λ)E 1 (w, W 1 ) − λH(w, x). (4.4.31) Table 4.1 summarizes different learning constraints with the corresponding h(w, x)-function. The bottom rows of Table 4.1 describe a few variations on the sum of entropy 8 The term h(w, x) can appear as a gradient of some information measure scaled by a positive definitematrix P(x, w), then in general corresponding H(w, x) may not exist. We use the negative sign beforeh(w, x) term in Eq. 4.4.30 for convenience, since most of the used feature extraction rules are formulatedas a maximization problem.
    • Chapter 4: Unsupervised learning constraints 62 Unsupervised Constraints Type of h(x, w) constraintsEntropy maximization (Bell and Sejnowski, 1995) with sigmoidal activation function:(ME) ∆W = η(I + (1 − 2y)ut )W ( 4.3.6 )BCM (Intrator and Cooper, 1992) with sigmoidal activation function ∆wij = ηE[φ(yi , θi )g (ui )xj ] ( 4.3.24 )Sum of entropies: ∆wij = η(f (ui ) + (1 − 2g(ui )))xj A f (u) = u3 B f (u) = 2tanh(u) C f (u) = 4 u + 15 u9 − 14 u7 − 29 u5 + 29 u3 3 11 4 3 4 4 D ∆wij = −η(f (ui ) + (1 − 2g(ui )))xj f (u) = [ 3 u11 + 15 u9 − 14 u7 − 29 u5 + 29 u3 ] 4 4 3 4 4Nonlinear PCA ∆W = ηy(xt − yt W)Table 4.1: Different learning rules used as unsupervised constraints in addition to recon-struction (see text and Appendix for details).rules, based on a different type of function f (u). These functions emphasize differentstatistical properties of the input distribution and are discussed in (Blais et al., 1998). Inparticular, the last two rows use the Gram-Charlier approximation to the entropy whichis done via moments (Stuart and Ord, 1994). The last row represents a minimization ofentropy rather than maximization, as might be suggested by the desire to find distributionsthat are far from Gaussian. Similar to the hybrid network with reconstruction constraints the constrained networkwith the learning rule (4.4.30) may be interpreted as a competitive learning of two nets forclassification and statistical feature extraction. The output layer of the feature extractionnetwork coincides with the hidden layer of the classification network. Thus, the hybridnetwork learns to classify and extract useful statistical properties simultaneously.4.5 Imposing unsupervised and reconstruction con- straintsGeneralizing our approach further, we offer to constrain classification by reconstructionand other types of unsupervised constraints (see Figure 3.4). The generalized learningrule has the form: ∆W1 = −η W1 E 1 (w, W1 ) ∆W2 = −η W2 E 2 (w, W2 )
    • Chapter 4: Unsupervised learning constraints 63 ∆w = −η ((1 − λ) w E 1 + λ((1 − µ) w E 2 − µh(w, x)), (4.5.32)where now we have two regularization parameters λ and µ. Thus, the most general net-work corresponds to the goal function (3.2.4) and its flow-chart is presented in Figure 3.4.
    • Chapter 4: Unsupervised learning constraints 64Appendix A to Chapter 4: Order statisticsHere we give some definitions and relations between order statistics (see (Stuart and Ord,1994)). Definition: Moments of order r about the point a ∞ µr = (x − a)r dF, (4.5.33) −∞where F is a distribution function. Definition: Characteristic function c.f. ∞ φ(t) = exp(itx)dF (4.5.34) −∞It may be easily seen that moments of distribution µr about pont 0 are related to ther-order derivative dr φ(t) of the characteristic function φ(t) via: t µr = (−i)r [dr φ(t)]t=0 t (4.5.35)Another set of statistical measures that are widely used in statistics are cumulants. Definition: The cumulants are defined by the identity ∞ kr (it)r /r! = log φ(t) (4.5.36) r=1Thus if a moment of order r µr is the coefficient of (it)r /r! in the Taylor series expansionof the characteristic function φ(t), kr is the coefficient of (it)r /r! in the Taylor seriesexpansion of log φ(t). Here we present the relation between the first four order statistics: k 1 = µ1 k2 = µ2 − µ12 k3 = µ3 − 3µ1 µ2 + 2µ13 k4 = µ4 − 4µ3 µ1 − 3µ22 + 12µ2 µ12 − 6µ14 In order to describe some interesting properties of the distribution, some other statis-tical measures have been defined: Definition: Kurtosis µ4 k4 kur(u) = 2 −3= 2 (4.5.37) µ2 k2The kurtosis characterizes the degree of peakedness of the graph of a statistical distri-bution. It is indicative of the concentration around the mean. Distribution for whichkurtosis is equal to zero is called mesocurtic. Those with positive kurtosis are called
    • Chapter 4: Unsupervised learning constraints 65leptokurtic and with negative platycurtic. Kurtosis is equal to zero for Gaussian distri-bution, is negative for sub-Gaussian and positive for super-Gaussian random variables.The super-Gaussian random variable is “sharper” than the Gaussian, its pdf has largetail areas and is more sharply peaked. The pdf of the sub-Gaussian random variable hassmaller tail areas and are also flatter-topped (see Figure 4.2). For a normally distributedrandom variable (µ1 = 0 and µ2 = 1), kurtosis coincides with the cumulant of the fourthorder. For a family of the density function: fα (x) = C1 exp(−C2 |x|α ), (4.5.38)where positive constants C1 , C2 are the normalization constants that ensure that fα is aprobability density of the unit variance: m2 1/2 m2 α/2 C1 = ( ) ; C2 = ( ) , m3 1 m1 where ∞ 2 1 m1 = exp(−|x|α )dx = Γ( ) (4.5.39) −∞ α α ∞ 2 3 m2 = x2 exp(−|x|α )dx = Γ( ) −∞ α αThe different values of the positive parameter of α exhibit different shapes of the distri- Pdf’s graphs for a family of the exponential density functions 1.4 1.2 1 α=0.75 0.8 0.6 α=2 0.4 α=5 0.2 0 −10 −8 −6 −4 −2 0 2 4 6 8 10Figure 4.2: Sample graphs for a family of the exponential density functions. This figuredemonstrates the typical shapes of the super-Gaussian (α = 0.75), and sub-Gaussian(α = 5) random variables.
    • Chapter 4: Unsupervised learning constraints 66bution. The random variable is super-Gaussian for 0.5 < α < 2 and is sub-Gaussian forα > 2 (Figure 4.2).Appendix B to Chapter 4: Derivation of the sum ofentropies learning ruleWe consider compression of the input x = (x1 , x2 , . . . , xn ) via the following nonlineartransformation: n ui = wij xj + wi0 , i = 1, . . . , m, m < n, yi = g(ui ). j=1u and y are vectors of pre and post-synaptic activations of the hidden layer, wij networkweights and wi0 network biases; and g-is a nonlinear monotone-increasing activation func-tion. The network architecture is presented in Figure 4.3. As a learning rule we choose Exploratory projection pursuit network Output y: y i =g i (u i ) yi ui u=Wx Input - xFigure 4.3: Feature extraction is achieved via (non linear) projection and dimensionalityreductionto maximize the sum of the entropies of the hidden units: m F(W) = H(yi ). i=1 The probability of the output of the hidden unit py (yi ) can be written as: pu (ui ) dyi py (yi ) = , where yi = yi duiThis leads to: p(yi ) ln(p(ui ))dyi = p(ui ) ln(p(ui ))dui ,
    • Chapter 4: Unsupervised learning constraints 67which implies the following expression for the sum of entropies: m m m yi F(W) = E[ ln ] = ( H(ui ))1 + E[( ln yi )2 ] i=1 p(ui ) i=1 i=1 Thus, our goal consists of two terms, the first , F1 is the sum of the entropies of the pre-synaptic activations of the hidden units and was evaluated by Amari et al. (1996) usingthe truncated Gram-Charlier expansion to approximate the probability density function(pdf) pu (ui ) and the second, E[F2 ] represents an expectation of the sum of the log-terms.The weights W have to be adjusted to maximize F(W). Using a gradient ascent algorithmwe obtain: ∂F1 ∂F2 ∆wij = η( + E[ ]) ∂wij ∂wijReplacing the gradient method by a stochastic method we obtain: ∂F2 ∆wij = η(f (ui )xj + ), ∂wijwhere f(u) is defined by Amari et al. (1996) and is the same as the function φ(ui ) in theexpression (4.3.14). However, in our simulation similar to Amari et al., we use f(u) as in(4.3.12,4.3.13). The second term for nonlinearities gi (ui ), may be written as: ∂F2 g = xt ∂W g g g1 (u1 ) g (um ) t =( ,..., m ) (4.5.40) g g1 (u1 ) gm (um )Thus, the learning rule simplifies to: g ∆W = η(f (u) + g )xt (4.5.41)The second term for any nonlinear function y = g(u), such that its derivative dependsonly on y itself yu = G(y), can be simplified by the following: ∂ln yi 1 ∂yi 1 ∂yi 1 ∂G(yi ) ∂G(yi ) = = xj = y i xj = xj ∂wij yi ∂wij yi ∂ui yi ∂yi ∂yi 1For the logistic sigmoidal activation function g(u) = 1+exp (−u) the derivative G(yi ) can beeasily evaluated as G(yi ) = yi = yi (1 − yi ). Thus, we obtain: ∆wij = η(f (ui ) + (1 − 2yi ))xj . (4.5.42)
    • Chapter 4: Unsupervised learning constraints 68 ∂F1 The only difference in the ∂wi j evaluation is the presence of the bias wi0 in (4.5.40).Therefore, we must require that the expectation of pre-synaptic activations of the hiddenunits ui be zero and their second moments be mi = E[u2 ] = 1. This can be achieved 2 iby normalizing u before the calculation of f (ui ). Furthermore, the network’s inputis normalized, so that E[x] = 0 and consequently, omitting bias at all (w0 = 0), thecondition E[u] = 0 is automatically satisfied. The second condition mi = E[u2 ] = 1 2 iconstrains the norm of W. The same rule, but with the negative η, can be used as agoal for sparse coding (Olshausen and Field, 1996; Atick, 1992).
    • Chapter 5Real world recognition5.1 IntroductionReal-world object recognition is impeded by natural climate conditions such as fog, rainor snow and also by other conditions such as partial occlusion and noise. This is fur-ther complicated by changes in illumination and shadows that are due to movement ofsurrounding objects. Some of these factors cause image blur, and all these factors arecrucial for recognition performance and have to be properly addressed during trainingand testing. This chapter addresses face recognition under various image degradations. We com-pare different regularized recognition networks and different ensembles by testing theirperformance on the degraded images. Results on two data-sets under various resolutionsand image degradations are demonstrated. We conclude that a combination that includesensembles with reconstruction constraints achieves the best performance on the degradedimages. In addition we show that via saliency maps reconstruction can deemphasize de-graded regions of the input, thus leading to classification improvement under “Salt andPepper” noise.5.1.1 Face recognitionFace recognition is an active field of research with possible applications in such areas asman-machine interaction, robotics, access control, automatic search in visual databasesand low bit-rate compression. This task is challenging, since faces do not appear asfixed image patterns; they can appear anywhere, at any size and orientation and withvaried background (Chellapa et al., 1995). Thus, face detection and normalization areusually performed that reduce variability caused by these factors. However, such local-ization preprocessing is not sufficient, since faces are not rigid and lighting conditions 69
    • Chapter 5: Real world recognition 70are not uniform. Different facial expressions, changes in hair-style and eyeglasses, andlighting conditions lead to a large amount of face variability. In some applications, thisnormalization may be further complicated by low quality of the images. For example,systems installed at airports yield foggy, blurred images; cheap cameras, such as thoseused for robot navigation, lead to images with low resolution. Thus, face recognition is aparticular case of the training when the variability of the data describing the same classis comparable with the similarity between different classes (Moses, 1994). Face recognition approaches can be divided into two basic groups, feature-based meth-ods (Samal and Iyengar, 1992, survey) and processing images as a whole (Kirby andSirovich, 1990; Turk and Pentland, 1991; Moghaddam and Pentland., 1994; Valentinet al., 1994, survey). Most of the effort in the feature-based methods is focused on findingindividual features (e.g., eyes, mouth, nose, head outline, etc.) and measuring statisticalparameters to describe those features and their relationship. Different methods for featureextraction were proposed such as template matching (Baron, 1981), deformable templates(Yuille et al., 1989), combination of perceptual organization and Bayesian networks (Yowand Cipolla, 1996) and methods using facial symmetry and elementary knowledge of faces(Reisfeld et al., 1990; Tankus, 1996), etc. However, selecting a set of features that cap-tures the information required for a face recognition is not easy and there is no a completesatisfactory solution to it. An alternative approach, inspired by the Gestalt school of perception (Hochberg, 1974;Kanizsa and Gaetano, 1970) is to process faces as a whole. One of the method presentingthis approach is PCA, that was used for face recognition (Kirby and Sirovich, 1990; Turkand Pentland, 1991, see Section 3.1.4 for description). Another way is to process imagesvia Neural Networks. Under this processing faces are presented as pixel intensity imagesand extraction of geometrical relationship, texture and subtle facial details is realizedimplicitly. Recognition from intensity images is also sensitive to substantial variationsin lighting conditions, head orientation and size. In order to avoid these problems, anautomatic preprocessing of the faces (i.e., normalization for size and position) is required.Although this normalization stage is also based on the feature extraction, it is ratherconstrained and completed by the definition of eyes and mouth or nose locations. Among the first network models proposed for face recognition are autoassociativenetworks and autoencoders (Valentin et al., 1994, survey). Although these network modelswere proposed for recognition, they are trained to reconstruct faces. In autoassociativenetworks, the recognition task is constrained to a face familiarity task. The cosine betweenevery face and its reconstructed version is evaluated and is thresholded to decide if the faceis familiar or not (O’Toole et al., 1991). In the autoencoders, their hidden representation
    • Chapter 5: Real world recognition 71has been used as an input for the back-propagation sex and identity networks withouthidden layer (Cotrrell and Fleming, 1990). Radial basis function (RBF) networks in the context of face recognition have beenfirst implemented by Edelman et al. (1992). The famous data-set (Turk and Pentland,1991) described below in Section 5.2.4 has been used in their experiments. The faceswere normalized by the same procedure, as described below in Section 5.2.5, to reducevariability to viewpoint and illumination direction. A set of Gaussian receptive fields(RFs) of different size and elongation were applied to reduce dimensionality of the input.These RFs were applied in different locations inspired by observation RFs of the simplecells in the primary visual cortex of mammals. Every RBF network was intended for acertain person recognition and was trained only by positive examples for which a singleoutput neuron had a desired value equal to 1. The face was considered as recognizedby the individual RBF network if its output exceeded some threshold. Later on, whentraining of the individual RBF networks was ended, their outputs were used as inputs toa new RBF network with the number of output units equal to the number of persons.The desired activities were taken to be equal to 1 for the neuron responsible for a giveninput image, and was equal to 0 for others. The misclassification rate equal to 9% vs.22% for individual networks was achieved by this new RBF network. Recently, an interest to RBF networks as a tool for face recognition has been revived.Different novel variants of the RBF network schemes were proposed (Howell, 1997; Satoet al., 1998; Gutta et al., 1996). In (Howell, 1997), the hyper RBF network, which hasthe number of hidden units equal to the number of training samples and trained on theimages of all persons, is reorganized into a group of smaller face recognition unit networks.Each face recognition unit network is intended for a particular person recognition and hastwo output units. The first unit is responsible for the particular person presence and thesecond has to be active when an ”anti” person is presented. The network uses views of thecertain person as positive examples and some selected ambiguous images of other peopleas negative ones. Although this approach increases complexity, as more networks need tobe trained, it allows to reduce dimensionality of each unit network and it is adaptive toa new person addition. When a new person is added, only one additional unit networkhas to be trained, and perhaps a small number of ambiguous unit networks needs to beretrained. A way to combine the standard RBF network with face unit networks basedon their confidences was also proposed. Ensembles of standard RBF networks for face recognition have been proposed in(Gutta et al., 1996). Two ensemble variants, defined in terms of their specific topol-ogy (connections and RBF nodes) and the data they are trained on, were considered.
    • Chapter 5: Real world recognition 72In the first variant (ERBF1), three groups of networks, which were trained separatelyon the original data, and on the same original data with either some Gaussian noise orsubject to some degree of geometrical distortion were combined. Inside each group threenetworks with the different topology were taken. The decision is based on the averagingof the networks outputs (see Section 2.2) and takes place if the maximal response is largerthan some threshold. In the second variant (ERBF2), three RBF networks with differenttopology were trained on the extended data consisting of original data and their corruptedversions. Later on these ensembles are combined with inductive decision trees classifiers. Sato et al. (1998) use as input to RBF networks partial face images, such as ears, eyesand nose, which are cropped by hand. The network is trained with sub-images of knownand unknown images, taken under uniform lighting condition and with the fixed distancebetween a camera and subjects. Each output unit of the RBF network corresponds tothe certain person. The input is recognized according to the unit with a maximal outputresponse, if the latter is larger than some threshold. This threshold is set by hand dueto separability of the maximal responses of known and unknown sub-images. Thus, anetwork is also able to reject unknown faces. A variant of a hybrid supervised/unsupervised network for automatic face recognitionhas been proposed by Intrator et al. (1996). A network is trained using a hybrid trainingmethod. This method is based on a formulation that combines unsupervised (exploratory)methods for finding structure (extracting features) and supervised methods for reducingclassification error. The unsupervised training is based on the biologically motivated BCMneuron (Intrator and Cooper, 1995) and is aimed at finding hidden units with a multi-modal distribution of their activities. The supervised portion is aimed at finding features(in network hidden units) that minimize classification error on the training set. The samedata-set and normalization as in (Edelman et al., 1992) were used. The classificationresult for averaged output of five hybrid BCM/recognition was 99.38%, which is betterthan using RBF networks (Edelman et al., 1992). A new approach to face recognition using Support Vector Machines (SVM) has beenproposed by Phillips (1998). SVM is a binary classification method that finds the optimallinear decision surface based on the concept of the structural risk minimization (Vapnik,1995). Since the face classification is a multi-class problem, the task has been previouslyreformulated as a two class recognition problem in a difference space (space of differencesbetween face images). In other words, the multi-class problem is replaced by the prob-lem of discriminating between within-class differences set (difference of faces of the samepersons) and between-class differences set (difference of faces of different persons). Theextension of SVM to nonlinear decision surfaces has been used and slightly adapted by
    • Chapter 5: Real world recognition 73introducing a threshold parameter ∆ to a decision surface parameterization. When thetask is recognition of some unknown probe face x, it is converted to a set of differencefaces x − xg , where xg are faces of known individuals, which are called a gallery set. Foreach difference face a similarity score δg , which depends on the decision surface parameters(but does not include ∆), is evaluated. The probe face is identified as a person for which aface xg from the gallery set has the minimal similarity score δg that satisfies the inequalityδg < ∆, otherwise the probe face is claimed as unfamiliar. When the probe is verifiedrather than identified the task is simplified, since the difference images are constructed asthe difference between the probe face and the faces of a person under verification. Someresults on the FERET database (Phillips et al., 1996; Phillips et al., 1997) are reported,such as a 77% − 78% classification rate. Although these results are not impressive, it ismarked that only two images per 50 different and the most difficult persons were used fortraining. Another approach to face recognition from live video has been recently proposed byAtick et al. (1997). Their scheme, called FaceIt, is based on the construction of thefactorial code, by transforming facial images into a large set of simpler statistically inde-pendent elements. The recognition task then consists of estimating the probability thata scene contains any pattern that was processed previously. Another different scheme which attempts to find a new good representation for facerecognition has been proposed in (Bartlett et al., 1998). Bartlett et al. used ICA (seeSection 4) for reduced face representation which was extracted using PCA. Classificationfrom the extracted independent components is improved compared to classification fromprincipal components. Another advanced feature-based method for face recognition using Hidden MarkovModels (HMM) has been proposed by Samaria et al. (1993). HMM models with thestates which are five facial features (forehead, eyes, nose, mouth and chin) are modeledand the HMM parameters are separately estimated for face images of the same person. Foran unknown face identification, its conditional probabilities given parameters of differentHMM models are evaluated and recognition is done as a label of the model with thehighest value of the conditional probability. Another advanced feature-based method is the dynamic link approach (Wiskott andvon der Malsburg, 1993; Wiskott et al., 1997). The method proceeds by applying Gaborfilters of 5 different frequencies and 8 orientations in a set of fiducial points (the pupils,the corners of the mouth, the tip of the nose, the top and bottom of the ears, etc.). Theobtained responses in every point compose the so called bunch Gabor jet. Subsequentlyevery known face is represented as a labeled graph of these fiducial points and edges
    • Chapter 5: Real world recognition 74between them. The nodes are labeled by their jets and edges are labeled with vectorsbetween the nodes, which they connect. The geometrical structure of the graphs unlabeledby jets is called a grid. It is assumed that different known faces have the same grids andcorrespondence between graph nodes of their models is set by hand. The face modelscorresponding to the same orientation are joined into FBG (face bunch graph), that hasthe average geometrical structure and combination of the bunch jets of all its models.Therefore, FBG is a representation of all faces with the same orientation. When an unknown image is given, its fiducial point locations which maximize a simi-larity between the unknown image graph and FBG are searched. The similarity measurebetween face graph and FBG is defined as a sum of jet and geometrical similarity measures,controlled by a trade-off parameter. The optimization task is simplified by constrainingthe group of possible geometrical transformation of FBG to translation, scale, aspect ra-tios and local distortions. Subsequently, the similarity measure between a found imagegraph and image graphs of all FBG faces are evaluated. Recognition is done picking up theknown face with the highest similarity measure. The similarity measure between imagegraphs is defined as the average similarity between corresponding jets. In this consider-ation, it is assumed that the unknown face is normalized, i.e. its position is estimatedbefore recognition procedure. In this chapter, we implement hybrid networks that were presented in Chapters 3–4 forface recognition. Our approach is the succession of the hybrid supervised/unsupervisednetwork approach (Intrator et al., 1996) with a novel type of unsupervised constraints.Different types of the bias constraints are given below in Section 5.2, where a regularizationprocedure is also presented. The regularization procedure is completed by creation ofvarious hybrid network ensembles. These ensembles are tested on degraded facial data-sets. Image degradation, which has been simulated in our experiments, is briefly describedin Section 5.3 and recognition results are presented in Section 5.4. In particular, for thesame data and normalization as in (Edelman et al., 1992; Intrator et al., 1996) (see alsoSection 5.2.5), we achieve a misclassification rate of 0.5% despite using smaller trainingand larger testing sets.5.2 MethodologyFace recognition problem requires extrapolation from the training set since its distributionmay be rather different from the distribution of the testing set. Thus this problem requiresan efficient use of a-priori knowledge that can be introduced in the form of bias constraintsduring training (Section 2.3).
    • Chapter 5: Real world recognition 755.2.1 Different architecture constraintsIn Chapter 3, reconstruction constraints were suggested as a learning bias and the hybridrecognition/reconstruction network was introduced (Figure 3.3). This hybrid networkattempts to improve the low dimensional representation by minimizing concurrently themean squared error (MSE) of reconstruction and classification outputs. The proposedreconstruction/classification network is controlled by a trade-off parameter λ and includesa conventional classification network for λ = 0. We refer to the networks correspondingto λ = 0 as unconstrained networks or conventional classification networks. In the specialcase of λ = 1, we get a nonlinear autoencoder for nonlinear activation functions and alinear autoencoder for linear (see Section 3.1.2). As has been discussed in Section 3.1.2,the linear autoencoder hidden weights span the PCA eigenspace. Below, we refer to thenetwork obtained in this case as a PCA network. All the networks corresponding to thetrade-off parameter inside the interval [0 1] are called the reconstruction networks. In Chapter 4, unsupervised constraints were introduced as statistical feature extractionconstraints on the hybrid network. The hybrid neural network with a hidden representa-tion that is arranged mainly for the classification task and, in addition, has some usefulproperties, was considered. We have used such statistical properties as an independenceof hidden neurons or maximum information transfer in the hidden layer. The proposedunsupervised/classification networks are also controlled by a trade-off parameter λ andinclude a conventional classification network for λ = 0. We consider several types ofunsupervised constraints (see also Table 4.1): • Entropy maximization constraint, which maximizes a joint entropy of the hidden layer (Section 4.3.1) • BCM constraints, which emphasize data multi-modality by minimizing a specific loss function (Section 4.3.4) • Sum of entropies of the hidden units constraints. We consider four variants of these constraints (see Table 4.1). Constraints A-C maximize the information carried by each hidden neuron (Section 4.3.5). The case D corresponds to the sum of entropies minimization. • Nonlinear PCA constraints, which extract nonlinear principal components in the hidden layer (Section 4.3.6) In the general case, bias constraints are a composition of reconstruction and unsupervisedconstraints (see Section 4.5). For simplicity we take these constraints with the same
    • Chapter 5: Real world recognition 76strength, i.e., the parameter µ in Eq. 4.5.32 is set to 0.5, and only the trade-off parameterλ is variable. In particular, we consider the combination of reconstruction and entropymaximization constraints. We refer to the corresponding hybrid networks as reconstruc-tion with entropy maximization networks. Thus, independent of the applied constraintsnetworks are controlled by a trade-off parameter λ and regularization is required.5.2.2 RegularizationRegularization task is to find an optimal parameter λ and corresponding synaptic weightsωλ which provide the minimal misclassification rate. The choice of the optimal parame-ter can be done by hold-out, cross-validation or bootstrap methods (see Appendix 2.8).We have not used cross-validation and bootstrap methods as they are computationallydemanding. Our regularization scheme is a variant of the split-sample validation method.We split the data into approximately equal portions of training and validation sets. Find-ing optimal weights ωλ depends on a stopping time in the training stage. The stoppingtime has been set observing the behavior of the misclassification rate on the validationset. Our regularization method includes the following steps (see Figures 5.1, 5.2, 5.3). 1. For every λ, train corresponding network until a minimum misclassification rate is achieved on the validation set within a predefined number of epochs . 2. Since the misclassification rate is a stepwise function, we further choose a stopping time, which corresponds to a minimum misclassification rate together with a minimal recognition MSE on the validation set. 3. A λ-value providing a minimum misclassification rate on the validation set is an optimal one. 4. Choose an ensemble of networks around the optimal λ value. Later this ensemble is combined with a zero-λ ensemble.In order to study solely effect of the trade-off parameter λ on the classification perfor-mance, we have fixed all other training conditions, such as initial weights and a learningrate. The initial weights have been chosen at random from a uniform distribution onthe interval [0, µ]. The learning rate has been taken small enough in order to ensureconvergence. From a practical viewpoint, the choice of the best network is not reasonable, sinceit depends on the degradation that is unknown a-priori. Instead of the search of theoptimal λ, we average over several regularization values, that is roughly equivalent to
    • Chapter 5: Real world recognition 77 Misclassification rate time evolution λ= 0 λ= 0.1 2 2 10 10 1 1 10 10 0 0 10 10 0 1000 2000 3000 4000 0 1000 2000 3000 4000 epochs epochs λ= 0.2 λ= 0.3 2 2 10 10 1 1 10 10 0 0 10 10 0 1000 2000 3000 4000 0 1000 2000 3000 4000 epochs epochsFigure 5.1: Validation set results vs. the regularization parameter λ. Regularization withλ > 0.3 provide larger error than with λ = 0.3 (see also the top graph of Figure 5.3).
    • Chapter 5: Real world recognition 78 MSE recognition error time evolution λ= 0 λ= 0.1 0 0 10 10 −1 −1 10 10 −2 −2 10 10 −3 −3 10 10 0 1000 2000 3000 4000 0 1000 2000 3000 4000 epochs epochs λ= 0.2 λ= 0.3 0 0 10 10 −1 −1 10 10 −2 −2 10 10 −3 −3 10 10 0 1000 2000 3000 4000 0 1000 2000 3000 4000 epochs epochsFigure 5.2: Validation set recognition MSE scaled per sample vs. the regularizationparameter λ.
    • Chapter 5: Real world recognition 79 error Classification based regularization. 7 6 Misclassification 5 4 3 0 0 0.1 0.2 0.3 −3 x 10 error 6.383 Recognition 4.417 0 0.1 0.2 0.3 λ error 0.0205 Reconstruction 0.0127 0 0.1 0.2 0.3 λFigure 5.3: Classification based regularization (for Pentland data-set in the intermediateresolution (32 × 32)): The upper graph shows the minimal number of misclassified faces inthe validation set versus λ. The middle graph shows a minimal mean-squared recognitionerror corresponding to the level of misclassification error in the upper graph. In thebottom graph the mean squared reconstruction error corresponding to the upper graphsis shown. All errors are calculated on the validation set per sample.
    • Chapter 5: Real world recognition 80integrating over a uniform regularizaion distribution between some values. Such averagingis equivalent to the Bayesian approach (see Section 3.2.3) for combining neural networkshaving the same evidences for the chosen interval of the hyper-parameter λ. We haveexperimentally found that training several networks on different λ values around severaloptimal values that were found once, and then averaging the different network results,yields a performance that is close to the optimal (a posteriori) λ and sometimes is evenbetter (see Section 5.4). Thus, we do not regard the need to estimate an appropriate λas problematic. In the results described below, we refer to an optimal λ as the one whichgives best test results versus degradation. It is thus clear that this is the upper limit ofperformance under this scheme and this limit can be attained and sometimes surpassedby a simple method of averaging over several λ values.5.2.3 Neural Network EnsemblesAn ensemble of experts is capable of improving the performance of single experts (Sec-tion 2.2). We have used two types of ensemble classification prediction. The first, is amajority rule over all the experts in the ensemble. We call this a classification ensemble.Another rule is based on averaging the real values of the outputs of all the ensemblemembers and then producing a decision by the Bayesian classification rule. We call thisa regression ensemble. It was shown (Section 2.2), that the largest reduction in the variance portion of theerror is achieved when the predictors are independent and this may be achieved by com-bining networks with different initial weights. We generate such ensemble of unconstrainednets (λ = 0) and use it as a baseline for ensemble performance comparison. It turns out that by averaging (in either way) over ensemble members that have beentrained with different values of the trade-off parameter λ (see Section 3.2.3), some ad-ditional independence is achieved, leading to a useful collective decision. We call theseensembles regularization ensembles and classify them further according to the trainingconstraints that were used during training of the ensemble networks. Therefore, ensem-ble with the networks constrained by the reconstruction task is called the reconstructionensemble, by BCM – the BCM ensemble, etc. Different ensembles are further combined with each other to generate more powerfulpredictors. The additional variance reduction is attained due to different constraints usedfor network training, that makes them independent. In particular, we have considered thecombination of the reconstruction and unconstrained λ = 0 ensembles, and the combina-tion of the reconstruction and reconstruction with entropy maximization ensembles. Werefer to the latter ensemble as the reconstruction and entropy maximization ensemble.
    • Chapter 5: Real world recognition 815.2.4 Face data-setsThe widely available facial data-set (Turk and Pentland, 1991) as well as a face data-setlocally collected by the Tel-Aviv University Computer Vision Group (Tankus, 1996) wereused in our simulations. While there have been many successful classification approachesto the Turk/Pentland data, we demonstrate that when the images are given in low res-olution, or are degraded either by blur or partial occlusion, classification performancedeteriorates dramatically. The Turk/Pentland data-set contains 27 images of 15 malefaces (we omitted the single bearded person). From each face, we randomly chose 14 train-ing images and 13 validation images (total of 210 training and 195 validation images).Preprocessing details and previous results studying the effect of background, illuminationand comparison with PCA are given in (Intrator et al., 1996). The preprocessing partiallyremoves the variability due to viewpoint, by setting (automatically) the eyes and tip ofthe mouth to the same position in all images (see Section 5.2.5). Further preprocess-ing evaluates the difference between each image and an average over all the training setpatterns, leading to the so called “caricature” images (Kirby and Sirovich, 1990). Threeresolutions were used: high - (64 × 64), intermediate - (32 × 32) and low - (16 × 16) pixels.Examples of a face in three resolutions are shown in Figure 5.4. “Caricature” faces in three resolutions Resolution 32*32 Resolution 64*64 Resolution 16*16 Figure 5.4: Pixel resolutions used in the classification results (Pentland data-set). The second data-set contains images of 37 male and female faces with 10 pictures foreach person in high resolution (84 × 56). We split the data to 6 training images and 4validation images for each person and used a similar preprocessing as described above,except that only the eye locations were fixed.5.2.5 Face normalizationThis section describes the face normalization which was used for the facial data-sets. Thenormalization is based on finding anchor points: eyes, nose or mouth and then warping theface images to some predefined locations of these points. The anchor points are identifiedusing the Generalized Symmetry Transform (Reisfeld, 1993; Tankus et al., 1997).
    • Chapter 5: Real world recognition 82 The method proceeds starting from an edge map and assigning a symmetry measureat each point, producing a ”symmetry map” of the image. A symmetry measure foreach point and direction is defined as follows. Let pk = (xk , yk ) be any image point and ∂I ∂I I(pk ) = ( ∂x , ∂y )|(x,y)=pk be the gradient of the intensity at point pk . The gradient isconsidered in the logarithmic scale, i.e. a vector vk = (rk , θk ) is associated with each point ∂I ∂Ipk , where rk = log(1+ I(pk ) ) and θk = arctan( ∂x / ∂y )|(x,y)=pk . For each two pointspi and pj the line l passing through them and the counterclockwise angle αij between itand horizontal are introduced. The set Γ1 (p, ψ), a distance weight function Dσ (i, j) anda phase weight function P (i, j) are defined by: Γ1 (p, ψ) = {(i, j)|(pi + pj )/2 = p, αi,j = ψ} Γσ (p) = {(i, j)|(pi + pj )/2 = p, pi − pj < 3σ} 1 pi − pj Dσ (i, j) = exp(− ) (2πσ) 2σ P (i, j) = (1 − cos(θi + θj − 2αij ))(1 − cos(θi − θj ))The first multiplier term of the measure Pij has peak when the gradients at pi and pjare oriented in the same direction towards each other, while the second term suppressesP (i, j) when θi = θj = π/2, which occurs for points lying in the straight line. The radialsymmetry measure M (p) and directional symmetry measure Sσ (p, ψ) of each point p indirection ψ are defined as : Sσ (p, ψ) = Dσ (i, j)P (i, j)r(i)r(j) (i,j)∈Γ1 (p,ψ) M (p) = Dσ (i, j)P (i, j)r(i)r(j)sin2 ((θi + θj )/2 − α(p)), where (i,j)∈Γ2 (p) α(p) = (θi + θj )/2 and (i , j ) = argmax(i,j)∈Γσ (p) Dσ (i, j)P (i, j)r(i)r(j)The maps produced by these operators are then subjected to detection of the highestpeaks. Geometrical relationship among these peaks, together with the location of themidline are defined to infer the face position as well as eyes and mouth in the image.Detection of the midline of the face image is found as a peak in the autocorrelationfunction of the edge map. Common information, such as the assumption that eyes shouldbe on both sides of the midline and the mouth should intersect it, is used.5.2.6 Learning parametersWe have used hidden layer consisting of 10 units for both data-sets. This number waschosen by trial. The value of the parameter µ which locates initial weights in the small
    • Chapter 5: Real world recognition 83vicinity of the weight space origin, was set to µ = 0.001 for the experiments with thePentland data-set in the intermediate resolution 32 × 32 and was set to µ = 4µ = 0.004and µ = 0.25µ = 0.00025 for low and high resolution, respectively, in order to obtain theconsistent results in all three resolutions. The number of predefined training steps was adhoc to 5000 epochs for intermediate resolution, 3000 epochs for high and low resolutions.The learning rate η has been adjusted according to the bias constraints. In experimentswith reconstruction constraints, the learning rate was equal 0.2. For the TAU data-set µwas equal 0.001, the learning rate η was set to 0.05 and number of epochs about 10000epochs was used.5.3 Type of image degradationsFor the Pentland data-set, we have performed experiments in three resolutions: low (16 ×16), intermediate (32×32) and high (64×64). The test images were obtained by simulatingdegradation on the validation-set only, i.e. all results are based on networks that weretrained on ”clean” data and were tested with either clean or degraded validation data. Afew examples of degraded faces and their reconstructed versions by different networks areshown in Figure 5.5. Below, we briefly describe the type of degradations that were used. For a comprehen-sive treatment of degradation see Chapter 6.“Clean” data: The original test set without any image degradation.Blurring with Gaussian filter: Blurring with a Gaussian filter is one of the simplesttypes of image degradations. We used a Gaussian blurring with a standard deviation σ =2. This scale of smoothing retains many details needed for human perceptual recognitionfor high resolution images, but for intermediate and low resolutions, many details aroundthe eyes and mouth appear to be lost.Blurring with DOG filter: Difference of Gaussians (DOG) filter, which produces aMexican hat type receptive field, is a form of image preprocessing known to be presentin early mammal vision (center-surround cells) (Marr, 1982; Kandel and Schwartz, 1991)(see also Section 6.2.1). Standard deviations of the on and off center (positive and negativeGaussians) were 1 and 2 respectively. This type of preprocessing is known to enhanceedges.
    • Chapter 5: Real world recognition 84 Image degradation and reconstruction (TAU data-set)Figure 5.5: Reconstruction is done using an architecture with reconstruction constraints.The faces in each row from left to right represent: A “clean” face, a corresponding “car-icature”, a degraded version, a reconstruction of the degraded version obtained by thefirst 10 Principal Components, a reconstruction by a single unconstrained Network withλ = 0; Reconstruction by a network ensemble with reconstruction constraints and tradeoffparameters λ = 0.04, 0.3.Degraded faces from top to bottom:Upper row: “Salt and Pepper” noise with 20% degradation. Middle row: nose area wasreplaced by average intensity in that area. Bottom row: DOG-blur with the deviation ofon and off center equal 1 and 3.Partial occlusion: This is achieved by replacing the pixel values at a certain rectan-gular area of arbitrary size in any part of the face by the average intensity of the pixelsin that rectangle.“Salt and Pepper” noise: This degradation replaces pixel intensities by either themaximum or minimum grey-level value at random locations of a certain percentage of theimage (Rosenfeld and Kak, 1982). Results presented here were done with 10% and 20%replacement.5.4 Experimental resultsTable 5.1 presents results on classification schemes generated by networks with recon-struction constraints and their combination into ensembles. The results are in three im-
    • Chapter 5: Real world recognition 85 Classification results for Pentland data-set Classification Low Intermediate High Classification Low Intermediate High scheme Resolution Resolution Resolution scheme Resolution Resolution Resolution 16 × 16 32 × 32 64 × 64 16 × 16 32 × 32 64 × 64 λ=0 3.1 2.6 1.5 λ=0 5.1 12.3 16.9 λopt 3.1 1.5 1.0 λopt 4.6 7.2 11.8 classification classification ensemble 2.6 0.5 1.0 ensemble 4.6 8.2 13.3 regression regression ensemble 2.6 1.0 0.5 ensemble 4.6 8.2 10.3 PCA 15.9 13.8 17.9 PCA 22.1 35.4 50.8 Classification Low Intermediate High Classification Low Intermediate High scheme Resolution Resolution Resolution scheme Resolution Resolution Resolution 16 × 16 32 × 32 64 × 64 16 × 16 32 × 32 64 × 64 λ=0 5.1 3.6 1.5 λ=0 36.4 13.8 5.6 λopt 3.6 2.6 1.5 λopt 34.4 11.7 3.6 classification classification ensemble 4.1 1.5 1.5 ensemble 33.8 14.9 4.1 regression regression ensemble 4.1 1.5 0.5 ensemble 32.3 13.3 2.6 PCA 16.4 14.8 17.9 PCA 46.7 33.3 26.7Table 5.1: Percent misclassification rate for Turk-Pentland data-set in three resolutions.Top left: on the “clean” testing set. Top right: Blurred images with a DOG-filter withσ1 = 1 σ2 = 2. Bottom left: Results of partial occlusion around the nose. Bottom right:Results of a “Salt and Pepper” noise of 20% of the image. For 32 × 32 resolution, singleunconstrained net with λ = 0 and reconstruction ensemble correspond to initial “weightsB” of Table 5.2. PCA stands for PCA network.age resolutions with different image degradations. They show that constrained networkswhich may not show significant performances difference on tests with original, undegradedtest-set, do show a significant improvement when tested with degraded images. Below,we highlight some consequences of Table 5.1.Single PCA network When λ = 1 and the activation functions of the hidden andoutput units are linear, the hidden weights of the network span the space of principaleigenvectors (Section 3.1.2). Classification results for network PCA representations arepresented in Table 5.1 (bottom rows). These results are inferior to other methods anddemonstrate that the first few principal components may be inefficient for classification1 .Ensemble combination Classification ensemble, or voting, is quite common in com-putational learning theory (Section 2.2). We find that a regression ensemble is superiorto classification ensemble especially in higher image resolutions. We note that for a re-gression ensemble variance reduction by averaging is achieved when the errors of thedifferent classifiers are independent. It appears that the use of different λ values leads to 1 It is known however, that a larger number of PCA produces improved results (Kirby and Sirovich,1990).
    • Chapter 5: Real world recognition 86some independence in misclassification and thus, the regression ensemble produces betterresults.Different image resolutions Generally, the results from the 16 × 16 resolution areonly slightly worse than results with higher resolutions. This resolution is less sensitive todifference of Gaussians blur, but very sensitive to “Salt and Pepper” noise which producessignificantly worse results. This is a strong indication to the usefulness of multi-resolutiondetection as a means to improve performance under various image degradations. In short, Table 5.1 indicates that reconstruction constraints under regression ensembleproduce more robust results. In the following set of experiments, we consider othernetwork constraints.5.4.1 Different architecture constraints and regularization en- semblesTable 5.2 presents results on different classification schemes that were generated by variousnetwork constraints and regression ensemble combinations. All results were obtained onthe Turk-Pentland data-set with an intermediate (32 × 32) resolution, using networkstrained with two sets of initial random weights A and B.Unconstrained ensembles The first two rows in Table 5.2 represent two single (con-ventional) unconstrained networks, corresponding to training with different initial weightsA and B. This serves as a base-line comparison and demonstrates the increased sensitivityof single networks to image degradation, in particular to blur. Before concentrating on theeffects of additional constraints we note that ensemble without additional constraints (1stnumbered row of Table 5.2) is already significantly better than a single network. Similarresults for the TAU data-set are presented in the second row of Table 5.3.Reconstruction networks and their ensembles The next two rows (numbered 2and 3, in Table 5.2) show the variability of the reconstruction ensemble results due toa different initial set of weights (A and B). Classification results of the reconstructionensemble for the TAU data-set are shown in the 3rd row of Table 5.3. For this (moredifficult) data-set, the ensemble of unconstrained networks is always inferior to the en-semble with reconstruction constraints. The largest difference between the unconstrainedand reconstruction ensembles is observed for blurred images. The 4th row of Table 5.2 represents the reconstruction ensemble composed from thenetworks of two reconstruction ensembles with weights A and B. The main observation
    • Chapter 5: Real world recognition 87 Different Ensemble Types (Pentland data-set) Ensembles: Optimal NN Regression Gaussian DOG Occlusion “Salt and Type of for ensemble filter filter nose half Pepper noise” regularization testing on testing σ1 = 1 area face d=0.1 d=0.2 constraints: set set σ=2 σ2 = 2 area Single unconstrained net with initial “weights A” 1.0 * 10.3 8.2 1.0 7.2 4.6 10.8 Single unconstrained net with initial “weights B” 2.6 * 12.8 12.3 3.6 8.7 6.2 13.8 1. Ensemble for λ = 0 and different initial weights 1.0 0.5 6.7 7.7 0.5 5.6 2.1 7.2 2. Reconstruction with initial “weights A” 2.1 2.1 8.2 4.1 1.5 5.6 6.7 12.8 λ : 0.05 0.1 0.3 0.35 3. Reconstruction with initial “weights B” 1.5 1.0 8.7 8.2 1.5 6.2 3.1 13.3 λ : 0.1-0.3, step 0.05 4. Reconstruction ensemble with initial “weights A,B” 1.5 1.5 6.2 4.6 2.1 4.6 4.1 9.7 5. Reconstruction (A+B) and λ = 0 ensembles 1.0 0.5 5.6 4.6 0.5 4.6 2.6 6.7 6. Reconstruction with entropy maximization 1.5 2.1 7.2 4.6 2.6 4.6 4.1 8.7 λ : 0.05 - 0.3, step 0.05 7. Reconstruction and entropy maximization ensembles 1.0 1.5 5.6 3.1 1.5 4.6 4.1 7.2 8. Entropy maximization λ : 0 - 0.4, step 0.05 0.5 1.5 8.7 4.1 2.1 6.7 3.6 11.3 9. BCM λ : 0.05 -0.3, step 0.05 2.1 2.6 11.3 5.1 2.1 8.2 4.1 11.8 10. Sum of entropies A λ : 0.05 0.1 0.2 0.25 0.3 1.5 2.1 8.2 7.7 2.6 3.6 4.1 10.8 11. Sum of entropies B λ : 0.05 0.1 0.2 0.25 0.3 1.5 2.1 8.2 7.7 2.6 3.6 3.6 13.3 12. Sum of entropies C λ : 0.05 0.1 0.2 0.25 0.3 0.5 2.1 7.7 7.7 2.1 6.7 2.1 9.7 13. Sum of entropies D λ : 0.05 - 0.3, step 0.05 1.0 2.1 8.2 6.7 2.1 5.6 3.6 8.7 14. Nonlinear PCA λ : 0-0.3, step 0.1 2.6 3.6 20 21.5 7.6 26.2 51.3 74.4Table 5.2: Percent classification error for different image corruptions on the Turk-Pentlanddata-set in intermediate (32 × 32) resolution. All results are for an ensemble of networksthat includes the indicated λ values. The column optimal NN refers to the single best inthe ensemble λ-network. In the Salt and Pepper experiments, either 10% or 20% of theimage were corrupted. Information that is not relevant for single networks is marked with*.
    • Chapter 5: Real world recognition 88is that this combined reconstruction ensemble is better than the unconstrained and re-construction ensembles with either weights A or B in classification of Gaussian and DOGblurred images. As can be seen the unconstrained (λ = 0) ensemble is slightly betterthan this combined reconstruction ensemble when distortion is small. This motivatedus to combine the unconstrained and the combined reconstruction ensemble (5th row ofTable 5.2). We note that this joined ensemble leads to robust results and is superior toother ensembles. Similar results for the TAU data-set, with reconstruction ensemble, arepresented in the 3rd and 4th rows of Table 5.3. Different Ensemble Types (TAU data-set) Type of Best single Regression Gaussian DOG Occlusion “Salt and regularization net results ensemble filter blur nose half Pepper noise” constraints: on test on test σ=2 with σ1 = 1 area face d=0.1 d=0.2 set set σ2 = 2 area Best single unconstrained net, λ = 0 15.5 * 23.0 44.6 29.1 35.1 24.3 31.1 Ensemble for λ = 0 and different initial weights 15.5 12.8 19.6 31.8 18.2 20.9 16.2 22.3 Reconstruction ensemble λ = 0, 0.04, 0.1, 0.2, 0.3 15.5 12.8 16.2 26.4 18.2 26.4 16.2 14.9 Joined reconstruction and unconstrained ensemble 15.5 13.5 16.9 23.6 16.9 20.9 14.2 16.2 Entropy maximization λ = 0 : 0.4, step 0.05 20.3 12.8 18.2 32.4 16.9 23 13.5 20.3Table 5.3: Misclassification error (in percent) for various ensembles and joined reconstruc-tion and unconstrained ensembles. Results are given for the TAU data-set with differentimage corruptions. In the “Salt and Pepper” experiments, 10% or 20% of the image werecorrupted. Information that is not relevant for single networks is marked with *.Ensembles with unsupervised constraints In a manner similar to networks withreconstruction constraints, we have generated different families of networks with unsu-pervised feature extraction constraints (see Section 5.2.1). The entropy maximizationconstraint is superior (on the TAU data-set) to an unconstrained ensemble under imageocclusions and all types of image degradations (the last row of Table 5.3). The BCMconstraint (9th row of Table 5.2) and the sum of entropies constraints were useful underimage degradations using a DOG filter.Ensembles with reconstruction and unsupervised constraintsThe reconstruction ensemble with entropy maximization (6th row of Table 5.2) is bet-ter than the unconstrained ensemble and the reconstruction constraint ensembles ”withweights B” in classification of DOG blurred images. Joining this ensemble with bothreconstruction ensembles attains results that are better than the joined ensemble with re-
    • Chapter 5: Real world recognition 89construction and no constraints ensemble for DOG blurred images (7th row of Table 5.2).In general, however, merging of ensembles with reconstruction constraints and with noconstraints (λ = 0) leads to more robust results and is superior to the joined reconstruc-tion and entropy constraints ensemble. Figure 5.6 summarizes most results of Table 5.2 and compares between different en-semble averaging schemes and different learning constraints on the original and degradedimages. It shows that the “joined reconstruction ensemble” (pink, fifth bar) performsbetter than each reconstruction ensemble from which it is composed. Additional mergingwith the unconstrained (λ = 0) ensemble (black, seventh bar), gives a better performancein most of the cases. The same useful property of the reconstruction ensemble can beobserved when merging reconstruction and entropy maximization ensembles (yellow, sixthbar). This ensemble is superior under DOG blurred images.5.5 Saliency detectionThis section presents a way to improve recognition of corrupted images using networkgeneralization ability to reconstruct. Due to the bottleneck structure of the network,reconstruction is efficient even when images contain a large amount of noise or are partiallyoccluded by contrast objects. Reconstructed images, which we call prototypes, are able torecover partially degraded or occluded areas of the input. However, there is a difficulty tofind these degraded areas in the input, or more generally, to define relevance (confidence)of the image areas. This task is common in artificial intelligence and robotic vision. It isreferred to as a saliency detection or intelligent filtering (Baluja, 1996). The definition ofrelevance depends on the desired task and the learning algorithm. For example, for an autonomous vehicle navigation (Baluja and Pomerleau, 1995) asaliency map derived from a specific Neural Network representation (see Section 3.1.5)was designed to highlight significant (salient) regions of the input and deemphasize unim-portant regions. Their saliency map is based on the difference between an input imageand its prediction by the network from the previous video frame. Below, we present asaliency map construction for still images that is suitable for the classification task anduses the hybrid network, which was introduced in Chapter 3.5.5.1 Saliency mapAfter training a difference map (image) xd given by the difference between the inputimage x and its prototype xp : xd = abs(x − xp ), can be used for extracting unreliableareas (areas with a large noise or unexpected objects) in the input image. Due to the
    • Chapter 5: Real world recognition 90 Summary of different networks and different image degradations a: "Clean" data b: DOG blurring 10.2564 Misclas. rate % Misclas. rate % 2.5641 1 1 Av Clas Reg Av Clas Reg c: "Salt and Pepper" noise d: Half face crop 17.4359 9.7436 Misclas. rate % Misclas. rate % 1 1 Av Clas Reg Av Clas RegFigure 5.6: Misclassification rate (%) for different network ensembles and different typesof image degradation for Pentland data-set 32 × 32: “Av”- average performance of net-work ensembles. “Clas”- performance of the classification ensembles. “Reg”- performanceof the regression ensembles. The bars in the group from left to right correspond to thefollowing ensembles of Neural Networks: 1. Ensemble of unconstrained Neural Networkswith λ = 0 and different initial weights. 2. Reconstruction with initial weights A. 3.Reconstruction with initial weights B. 4. Reconstruction with entropy maximization. 5.Joined reconstruction ensemble (merged 2 and 3 ensembles). 6. Joined reconstruction andentropy maximization ensemble (merged 2,3 and 4 ensembles). 7. Joined reconstructionand unconstrained λ = 0 ensemble (merged 1,2 and 3 ensembles). For comparison, clas-sification errors of single Neural Networks with initial weights A are shown by horizontallines of dashed type.
    • Chapter 5: Real world recognition 91bottleneck structure of the network, the output of the reconstruction layer has to be betterfor recognition than the original signal, in areas where xd is large, i.e., the original signalx is messy. Thus, we propose before recognition to replace the original image x by theimage xn using a saliency map Φ(xd ): xn = Φ(xd )x + (1 − Φ(xd ))xp , (5.5.1)where all operations are pixel-wise. We have constrained a saliency map Φ(xd ) to be a decreasing function, such thatΦ(0) = 1 and have considered two types of saliency maps. The first type of saliency mapsis given by: Φ(x) = exp(−µx2 ) and parameter µ, tuned to µ = 0.9. The second saliencymap was taken as: 1 if x < x ¯ Φ(x) = 0.5 otherwisewhere a threshold x was adjusted to 0.3. Figure 5.7 shows examples of the xn images ¯obtained using two saliency maps. Classification was improved for some types of the Saliency map construction Input Reconstruction Difference Saliency Reconstruction Saliency Reconstruction map-1 with map-1 map-2 with map-2Figure 5.7: Reconstruction using saliency maps for network with reconstruction con-straints and trade-off parameter λ = 0.04 (TAU data-set). The white pixels of the firstmap (map-1) correspond to intensity equal to 1 and black to zero intensity. In map-2 thewhite pixels have intensity equal to 0.5 and black 0.degradation process, especially for “Salt and Pepper” noise (Tables 5.4–5.5). For other
    • Chapter 5: Real world recognition 92 Recognition using saliency maps (Pentland data-set) Types of Regression Ensembles degradation Unconstrained Reconstruction A Reconstruction B Joined “Salt and Pepper” noise with d = 0.1 input 1.5 2.1 3.1 1.0 map-1 1.5 3.1 3.1 1.0 map-2 1.5 3.1 4.1 1.5 “Salt and Pepper” noise with d = 0.2 input 7.2 11.3 11.3 6.7 map-1 2.6 4.6 4.1 3.6 map-2 2.6 4.6 5.6 3.6 “Salt and Pepper” noise with d = 0.3 input 25.1 26.2 30.8 23.6 map-1 11.3 13.3 13.3 11.8 map-2 12.8 14.9 15.9 12.3 “Right eye” with ν = 3 input 3.1 1.5 2.1 2.1 map-1 3.1 1.5 2.1 0.5 map-2 2.6 2.1 2.6 1.0 “Half face” with ν = 3 input 15.9 26.2 22.1 18.5 map-1 16.9 26.2 22.1 16.9 map-2 16.9 25.6 19.5 17.4 “DoG 1-2” input 7.7 4.1 9.2 4.6 map-1 7.7 3.6 9.7 4.1 map-2 8.2 3.6 8.7 5.6Table 5.4: Percent misclassification error results for images obtained using two types ofsaliency maps. Reconstruction ensembles A and B correspond to training with weightsA and B. In contrast with reconstruction ensembles A–B in Table 5.2 (2-3 rows), theycontain also one unconstrained network (λ = 0). Rows marked with “input” show standfor the input degraded images. Pentland data-set at 32 × 32 resolution.
    • Chapter 5: Real world recognition 93 Recognition using saliency maps (TAU data-set) Types of Regression Ensembles degradation Unconstrained Reconstruction Joined “Salt and Pepper” noise with d = 0.1 input 16.2 16.9 13.5 prototype 23.0 16.2 14.9 map-1 14.2 14.2 12.8 map-2 14.2 14.9 12.2 “Salt and Pepper” noise with d = 0.2 input 25.0 20.3 20.3 prototype 31.1 18.2 20.3 map-1 18.2 14.2 12.8 map-2 18.2 15.5 12.8 “Salt and Pepper” noise with d = 0.3 input 37.8 31.8 31.1 prototype 43.2 31.1 32.4 map-1 25.7 18.9 20.3 map-2 21.6 17.6 18.9 “Right eye” with ν = 3 input 14.2 15.5 13.5 prototype 14.9 14.9 13.5 map-1 14.2 15.5 13.5 map-2 13.5 15.5 13.5 “Half face” with ν = 3 input 43.9 36.5 34.5 prototype 43.2 41.9 36.5 map-1 42.6 36.5 34.5 map-2 41.9 35.8 36.5 “DoG 1-2” input 31.8 26.4 23.6 prototype 33.1 26.4 27.0 map-1 32.4 27.7 24.3 map-2 33.1 27.0 25.0Table 5.5: Percent misclassification error results for images obtained using two types ofsaliency maps. Rows marked by “prototype” stand for the reconstructed images (TAUdata-set).
    • Chapter 5: Real world recognition 94types of image degradation, classification improvement was not significant. To enforcethe efficiency of the saliency map, in the experiments with partially occluded images, theoccluded region was enhanced by multiplying the average intensity over the occluded areaby some factor ν.5.6 ConclusionsWe have shown that constraints on the properties of the low-dimensional internal repre-sentation of the images, such as entropy maximization, BCM and the sum of entropies,are useful and can be considered in conjunction with reconstruction constraints, to im-prove generalization for classification. It was further shown that an averaging of NeuralNetworks with different constraint strengths is preferable to a simple choice of the optimalregularized network parameters. The best classification results were obtained by mergingthe ensemble with reconstruction constraints and the unconstrained, λ = 0 ensemble. Reconstruction constraints significantly improve classification results under partialocclusion, lossy compression, “Salt and Pepper” noise and some image blur operations. Inaddition, we have shown that via saliency maps, reconstruction can deemphasize degradedregions of the input, thus leading to classification improvement under “Salt and Pepper”noise. In the next chapter, we investigate the influence of the reconstruction constraintson image recognition under a wide family of image blur and consequent deblur operations.
    • Chapter 5: Real world recognition 955.7 Appendix to Chapter 5: Hidden representation explorationImage recognition improvement is based on the extraction of a good hidden data rep-resentation. Although recognition performance is a single reliable measure that allowsone to judge the hidden representation quality, it may be interesting to consider somestatistics of the hidden layer units. Statistics of the hidden unit activities characterizethe data distribution after projection on the hidden weight directions. Some propertiesof the hidden representation are presented below. In Figures 5.8–5.9, the hidden unit activities per classes and different bias constraintsare shown. As can be seen, in both networks images of the same class excite similaractivation patterns in their hidden space and at the same time there is a big differencebetween patterns corresponding to different classes. It is clear that such a representationhas to be good for recognition. However, from the observations it is difficult to decidewhich type of constraints is preferable. The pdfs of the hidden unit activities are presented in Figure 5.10. As can be seen,they are multi-modal for unconstrained network and multi-modal or super-Gaussian forreconstruction network. Both these properties are useful for recognition (Chapter 4). Another way to get some impression about hidden layer structure is to look at thehidden weights as images (Figure 5.11). We note, however, that network ensemble hiddenrepresentation is not well defined.
    • Chapter 5: Real world recognition 96 Hidden unit activities vs. classes - for an unconstrained network Unconstrained network λ = 0 2 2 2 Neuron Neuron Neuron 4 4 4 6 6 6 8 8 8 10 10 10 class−1 class−2 class−3 2 2 2 Neuron Neuron Neuron 4 4 4 6 6 6 8 8 8 10 10 10 class−4 class−5 class−6 2 2 2 Neuron Neuron Neuron 4 4 4 6 6 6 8 8 8 10 10 10 class−7 class−8 class−9 2 2 2 Neuron Neuron Neuron 4 4 4 6 6 6 8 8 8 10 10 10 class−10 class−11 class−12 2 2 2 Neuron Neuron Neuron 4 4 4 6 6 6 8 8 8 10 10 10 class−13 class−14 class−15Figure 5.8: Results on “clean” Pentland data set at intermediate resolution 32 × 32. Eachsquare area represents a neuron activity magnitude vs. different inputs (such representa-tion is similar to Hinton diagrams for network weights representation). The color indicatesa magnitude sign: red for negative and green for positive values (in non colored printers,a red color appears more dusk than a green color).
    • Chapter 5: Real world recognition 97 Hidden unit activities vs. classes - for a reconstruction network Reconstruction network λ = 0.3 2 2 2 Neuron Neuron Neuron 4 4 4 6 6 6 8 8 8 10 10 10 class−1 class−2 class−3 2 2 2 Neuron Neuron Neuron 4 4 4 6 6 6 8 8 8 10 10 10 class−4 class−5 class−6 2 2 2 Neuron Neuron Neuron 4 4 4 6 6 6 8 8 8 10 10 10 class−7 class−8 class−9 2 2 2 Neuron Neuron Neuron 4 4 4 6 6 6 8 8 8 10 10 10 class−10 class−11 class−12 2 2 2 Neuron Neuron Neuron 4 4 4 6 6 6 8 8 8 10 10 10 class−13 class−14 class−15 Figure 5.9: Results on “clean” Pentland data set at intermediate resolution 32 × 32.
    • Chapter 5: Real world recognition 98 Pdf’s of hidden unit activities Unconstrained network λ = 0 neuron−1 neuron−2 neuron−3 neuron−4 neuron−5 neuron−6 neuron−7 neuron−8 neuron−9 neuron−10 Reconstruction network λ = 0.3 neuron−1 neuron−2 neuron−3 neuron−4 neuron−5 neuron−6 neuron−7 neuron−8 neuron−9 neuron−10Figure 5.10: Hidden unit activity pdfs - for unconstrained λ = 0 and reconstructionnetwork λ = 0.3 for “clean” Pentland data set at intermediate resolution 32 × 32.
    • Chapter 5: Real world recognition 99 Hidden weight representation Unconstrained network λ = 0 Reconstruction network λ = 0.3 Figure 5.11: Pentland data set at intermediate resolution 32 × 32.
    • Chapter 6Blurred image recognitionThis chapter studies a case where the required generalizations are for data which maybe “far” from data in the training set, namely data with a different distribution thanthe training set. In the previous chapter, we considered unsupervised and particularlyreconstruction constraints, as a mechanism to impose useful bias during training. Wehave shown that these constraints improve generalization performance for various imagedegradations, such as “Salt and Pepper” noise, low resolution and partial occlusion. How-ever, sensitivity to image blur was still too high. This chapter is devoted to performanceimprovement under various types of image blur.6.1 MethodologyRecognition of blurred images requires a substantial amount of training data processedby different blur operators. Unfortunately, such data is not available, and therefore, analternative way to solve the problem is to impose a priori information about possibledegradation transformations. For example, in the character recognition problem, the possible transformations aregeometrical, such as shift, rotation and scaling (Simard et al., 1992; Baird, 1990). Theregularization there appears as the invariance tangent prop constraints in the form ofthe penalty term to the cost function or using the distortion model, i.e. by data drivenregularization (Section 2.3.2). We choose to add Gaussian blurred images to the training set as a representative of allblur operations and recognition is done on a wide variety of blur operations. We furtherpropose to enforce reconstruction of blurred images to either their copy or to the originalnon-blurred images. Such training causes the hidden representation to become insensitiveto blur operation. Another obvious way to improve classification of the blurred images is to restore the 100
    • Chapter 6: Recognition of blurred images 101blurred images beforehand. In this case, before testing the recognition system on blurredimages, their degradation is reduced via image restoration techniques.6.1.1 Experimental designTraining schemesIn Chapter 5, hybrid networks were trained to classify and reconstruct “clean” images(Figure 6.1 A, training stage), i.e., the reconstruction of a copy of the input in the outputlayer was used. Below, we refer to this training scheme as training scheme A. Thistraining encourages internal representation where patterns of the same class are clusteredtogether (due to the reconstruction part of the learning), while the distance betweenpatterns of different classes is stretched (due to the discriminative/classification part oflearning) (Gluck and Myers, 1993). As a result, classification in this hidden space issimpler and is more robust to various forms of degradation. To further improve recognition of degraded images, we have added Gaussian blurredimages (with standard deviation σ = 2) during training. This data expansion proceduregives two additional types of the training procedure with reconstruction constraints. Thefirst training scheme B enforces reconstruction of the original “clean” images from theblurred inputs (Figure 6.1 B, training stage) and the second scheme C, is a simple dupli-cation of the inputs at the output (Figure 6.2). Both training schemes B and C encourageinternal representation to be more robust to blurring, but training scheme B introducesadditional invariance constraints on the image reconstruction task. As in Chapter 5, three types of ensembles are studied for each of the training schemesA–C: unconstrained, with reconstruction constraints and joined. The number of networksin the unconstrained ensemble of all schemes A–C is equal to 6. Ensembles with recon-strcution constraints of all schemes A–C have been composed from networks with thetrade-off parameter λ, which changes from 0 till 0.3 with an increment of 0.05.Testing schemesTwo testing schemes were used to evaluate the generalization ability of networks andtheir ensembles. The first testing scheme A is the same as in Chapter 5, i.e., variousimage degradations are simulated and a misclassification rate for different ensembles isevaluated (Figure 6.1 A, testing stage). In testing scheme B, the degraded images arefirst preprocessed using several restoration methods and only then classification is carriedout (Figure 6.1 B, testing stage). Our experiments consist of several groups which differby simulated degradation types and applied restoration techniques. In the next section,
    • Chapter 6: Recognition of blurred images 102 Experimental design schemes A B Training stage Training stage Reconstruction Reconstruction clean image clean image clean image Classification Class label clean image Class label Reconstruction Classification Blurred image clean image A B Testing stage Testing stage Class Class Restoration ? ? Blurred image Degraded image Restored imageFigure 6.1: (A): In the training stage, networks are trained to classify and reconstruct“clean” images. In the testing stage A, generalization ability to classify artificially de-graded images is tested; (B): Artificially blurred images are added to the training stage.Networks are trained to classify images and reconstruct their “clean” prototypes. In thetesting stage B, restoration preprocessing is used before recognition schemes.we review image degradation operations and restoration methods which we apply.6.2 Image degradationUsually degradation process is modeled as both a space-invariant blurring with a convo-lution operator h and a corruption with an additive noise n: g = h ∗ f + n, (6.2.1)where f is the original image. The major known causes for image blur are misfocus, camerajitter, object motion and atmospheric turbulence. These types of blur lead to a low passoperation on the image. Of particular interest is a difference of Gaussians (DOG) filter,which is a band-pass filter, and is known to be present in early mammal vision (Kandeland Schwartz, 1991). This operator is equivalent to simultaneous image smoothing andenhancement. A third family of image filters is the high pass filter which leads to imagesharpening. This filter is common in medical imaging, industrial inspection and military
    • Chapter 6: Recognition of blurred images 103 Training scheme C Training stage Reconstruction clean image clean image Classification Class label Reconstruction Blurred image clean imageFigure 6.2: In the training stage, the network is trained to classify and reconstruct “clean”and blurred images. Reconstruction is a copy of the input in the output sublayer.applications. The presence of noise in images is inevitable. It may be a result of imagegeneration, recording, transmission, etc. Noise corruption complicates image acquisitionand even a small amount of it is harmful for restoration of blurred images. We considertwo types of additive noise: Gaussian white noise and pulse noise. We limit ourselves toGaussian noise that acts independently on each pixel, with zero mean and some variance σ.Pulse noise (otherwise called “Salt and Pepper” noise) replaces pixel intensities by eitherthe maximum or minimum grey-level values with some probability (Rosenfeld and Kak,1982), producing separate high contrast black-and-white points. This noise is common invideo transmission.6.2.1 Main filtersFiltering may be done both in the frequency and spatial domains. Convolution in thespatial domain is equivalent to multiplication of the Fourier transforms of the image andthe filter in the frequency domain. In each particular case we indicate in which domainfiltering is done and represent point spread function or its Fourier transform (referred toas a transfer function) as required. Examples of images with various degradations areshown in Figure 6.3.Ideal filtersIdeal filters represent a class of frequency domain filtering that are easy to simulate.Transfer functions of these filters are radially symmetric about the origin and thoughthey are not physically realizable, they are widely used in image processing for comparingthe behavior of different types of filters. The name ideal indicates that some specified
    • Chapter 6: Recognition of blurred images 104 Degraded Images original a b c d e f g h iFigure 6.3: a) Result of Gaussian noise with σ = 2; b) Result of pulse noise with density20%; c) Result of replacement of the nose area with average intensity over this area; d)Result of the root filter with α = 0.6; e) Result of the out-of-focus filter with the blurradius R = 5; f) Motion blur with blur propagation on 7 pixels; g) Result of Gaussianblur with σ = 2; h) Result of the DOG filter with on and off centers equal to σ1 = 1 andσ2 = 2 i) Result of the ideal high pass filter with cutoff w = 3frequencies are completely eliminated. Depending on the eliminated frequencies ideal low,band and high pass filters are known (Gonzalez and Wintz, 1993). A transfer function ofthe ideal filter in the frequency domain (u, v) is given by the expression: 1 if (u, v) ∈ D H(u, v) = 0 otherwise,where the area of the unchanged frequencies D is: √ √ √ a) u2 + v 2 ≤ W , b) u2 + v 2 ≥ W 0 , c) W < u2 + v 2 < W 0 ,for low, high and band-pass filters respectively, W , W 0 are called cutoff frequencies.Motion blurMotion blur is a form of image degradation that may degrade recognition performance(Figure 6.3f). It is due to a relative motion between the camera and the object. Assumingthat a relative camera motion is horizontal and uniform and the total displacement duringthe exposure time T is a, the transfer function H(u, v) (Gonzalez and Wintz, 1993) isgiven by: T H(u, v) = sin(πua) exp(−πiua). (6.2.2) πua
    • Chapter 6: Recognition of blurred images 105H vanishes at values of u given by u = n , where n is a nonzero integer. In general, the aamplitude of H(u, v) is characterized by periodic lines of zeros, which are orthogonal to 1the direction of motion and are spaced at intervals of a in both sides of the frequencyplane.Out-of-focus blurThe point spread function (PSF) of a defocused lens with a circular aperture is approxi-mated by the cylinder whose radius R depends on the extent of the focus defect (Cannon,1976): 1 πR2 if x2 + y 2 ≤ R2 h(x, y) = 0 otherwise,where R is the “blur radius” which is proportional to the extent of defocusing. The Fouriertransform of h(x, y) in this case is H(u, v) = J1 (πRr)/(πRr), where J1 is the first-orderBessel function and is characterized by “almost-periodic” circles with zero valued H(u, v).This occurs for r satisfying: 2πRr = 3.83, 7.02, 10.2, 13.3, 16.5 . . . The well-defined structure of H(u, v) zeros in the case of motion and misfocus blur isused for the identification of the blur parameter (Cannon, 1976; Fabian and Malah, 1991)for the purpose of image restoration. However, these methods are sensitive to noise. Toovercome this drawback, some preprocessing stage for noise reduction and estimation wereused (Fabian and Malah, 1991). An example of a misfocus image with blur radius R = 5is shown in Figure 6.3e.Gaussian blurGaussian blur may be caused by atmospheric and optical blur. It is known that theeyes’ lenses cause such blur. Computer tomography images also suffer from Gaussian blur(Kimia and Zucker, 1993). The Gaussian convolution filter written in polar coordinatesh(r, φ) in the spatial domain is given by: −r2 h(r, φ) = Cσ −2 exp( ), (6.2.3) 2σ 2where C is a normalization constant. The lack of zero crossing of the Gaussian filter in thefrequency domain makes its identification very difficult. Moreover, Gaussian deblurringis numerically unstable (Humel et al., 1987; Kimia and Zucker, 1993). An example of animage blurred by this filter with σ = 2 is shown in Figure 6.3g.
    • Chapter 6: Recognition of blurred images 106DOG filterThe difference of Gaussian (DOG) filter is a good approximation to the circular symmetricMexican hat type receptive fields (center-surround) found in early mammal vision (Marr,1982; Kandel and Schwartz, 1991). It performs a band-pass filter that is the result ofapplying the Laplacian operator 2 to an image which is blurred with a Gaussian filter.The zero-crossings of the resulting convolved image are commonly used for edge detectionand segmentation. The DOG filter written in polar coordinates is described by: −2 −r2 −2 −r2 h(r, φ) = Cσ1 exp( 2 ) − Cσ2 exp( 2 ), (6.2.4) 2σ1 2σ2where σ1 < σ2 and are the standard deviations of the on and off center (positive andnegative Gaussians). An image blurred with a DOG filter is shown in Figure 6.3h.Root filterRoot filter is commonly used for image enhancement and deblurring (Jain, 1989). It affects ˆthe magnitude of the frequency response of an image V as given by: V = V α . Forsmall values of α < 1, it acts as a high pass filter, increasing the ratio between amplitudesin the high and low frequencies. An image enhanced with a root filter (α = 0.6) is shownin Figure 6.3d.6.2.2 Other types of degradationNoiseWe consider two types of additive noise: Gaussian white noise and pulse noise. Gaussianwhite noise is commonly used to model sensor noise and quantization process. We limitourselves to Gaussian noise that acts independently on each pixel with zero mean andsome variance σ 2 (Figure 6.3a). Pulse noise replaces pixel intensities by either themaximum or minimum grey-level value with some probability (Rosenfeld and Kak, 1982),producing separate high contrast black-and-white points. This explains why pulse noiseis called otherwise ”Salt and Pepper” noise. Pulse noise often appears during TV imagetransmission (Figure 6.3b).OcclusionOcclusion occurs as a result of motion, when two or more objects touch or overlap oneanother. Another cause for occlusion in 2D images is the change of viewpoint, whenpart of an object is occluded by another one. We simulate occlusion by replacing pixel
    • Chapter 6: Recognition of blurred images 107intensities at a certain rectangular area in any part of the image by some constant intensityin that rectangle (Figure 6.3c). A level of occlusion is characterized by a factor ν to theaverage intensity of an occluded area.6.3 Image restorationImage restoration refers to the problem of recovering an image from its blurred and noisyversion, using some a priori knowledge of the degradation phenomenon and the imagenature. It is well-known that the restoration problem is an ill-posed problem (Gonzalezand Wintz, 1993; Jain, 1989; Stark, 1987), i.e. a small noise in the observed image resultsin an unbounded perturbation in the solution. This instability is often addressed bya regularization approach (Tikhonov and Arsenin, 1977; Katsaggelos, 1989; Sezan andTekalp, 1990; Rudin et al., 1992; You and Kaveh, 1996) that includes restricting the setof admissible solutions and introducing some a priori knowledge about the image and thedegradation model.6.3.1 MSE minimization and regularizationAssuming the blur operator H is known, a natural criterion for estimating an originalpixel image f from an observed pixel image g in the absence of any knowledge aboutnoise, is to minimize the difference between the observed image and a blurred version ofthe restored image: 2 min M(f ) = min g − Hf . (6.3.5) f fOften, gradient or conjugate gradient descent methods are used for M(f ) minimization(Katsaggelos, 1989; Sezan and Tekalp, 1990). An application of the gradient method tothe minimization problem (6.3.5) produces the following iterative scheme: fk+1 = fk + β(Ht g − Ht Hfk ), f0 = 0. (6.3.6)When the blur matrix H is nonsingular and β is sufficiently small, the iterative scheme ˆconverges to the f = H −1 g. This solution is known as the inverse filter method. In thefrequency domain, it corresponds to the following estimation of the ideal image frequencyresponse: ˆ G(u, v) F (u, v) = . (6.3.7) H(u, v)As mentioned before, blur such as motion or defocusing leads to a singular H matrix. Inthis case, the above optimization method yields an iterative scheme that converges to the
    • Chapter 6: Recognition of blurred images 108minimum norm least square solution H + g of Eq. 6.3.5 (Katsaggelos, 1989; Jain, 1989),where H+ is the generalized inverse of matrix H. In the presence of noise the iterative algorithm converges to H + gb + H + n (where gbis a blurred image without noise interference) and thus contains noise filtered by thepseudo-inverse matrix. Often, H is a low-pass filter, therefore, the noise is amplified andthe obtained solution may be very far from the desired one. To overcome this sensitivity to noise, some a priori information about the noise orthe ideal image is often introduced as a quantitative constraint that replaces an ill-posedproblem by a well-posed one. This method is called regularization. The most well knownregularization methods (Tikhonov and Arsenin, 1977; Sezan and Tekalp, 1990) have ageneral formulation as a minimization of the function: 2 2 L(f ) = Hf − g +α Cf ,where the regularization operator C is chosen to suppress the energy of the restored imagein the high frequencies, that is equivalent to an assumption about the smoothness of theoriginal image in the spatial domain. Since usually the H filter is a low pass filter, inorder to obtain the smooth original image, the regularization operator C is taken to bea Laplacian · f , where – is a differential operator. A regularization parameter αmay be known a priori or estimated, but theoretically it is inversely proportional to thesignal to noise ratio (SNR). Although regularization of the MSE criterion with smoothness constraint Cf isthe basis for most of the work in image restoration, it often leads to unacceptable ringingartifacts around sharp intensity transitions. This effect is due to image blurring aroundlines and edges. Some solution to this problem is given by the following functional mini-mization (Katsaggelos, 1989): L(f ) = [g(x) − h(x) ∗ f (x)]2 + λ ω(x)[c(x) ∗ f (x)]2 . (6.3.8) x∈Ω x∈ΩThe first term in (6.3.8) represents the fidelity of the restored image with respect toan observation and the second represents a smoothness constraint, ∗ – is a convolutionoperator. The space adaptivity is achieved through the introduction of the weight functionω. The weight function ω is set to be small around the edge areas, larger near the smoothareas and usually is taken in practice as the inverse of the local variance of the image. The space adaptivity approach has been extended to the case of an unknown bluroperator (You and Kaveh, 1996; Chan and Wong, 1997). The method incorporates apriori knowledge about the image and the point spread function (PSF) simultaneously. Itproceeds by minimizing the cost function, which consists of a restoration error measure
    • Chapter 6: Recognition of blurred images 109and two regularization terms for the image and the blurring kernel; under constraints onthe blur filter energy. You et al. (You and Kaveh, 1996) formulate the problem as a minimization of thefunction dependent on the discrete image and filter values (2D image and filter functionsare quantized on the grid): L(f, h) = ω(x)[g(x) − h(x) ∗ f (x)]2 + x∈Ω λ1 ω1 (x)[c1 (x) ∗ f (x)]2 + λ2 ω2 (x)[c2 (x) ∗ h(x)]2 (6.3.9) x∈Ω x∈ΩIn (6.3.9) the first term is responsible for the image fidelity and the second and third termsrepresent smoothing constraints on the image and the blur filter, respectively. Smoothnessis introduced adaptively via the weights ω1 (x) and ω2 (x). Though the gradient descent method is commonly applied for minimization, an al-ternating minimization (AM) algorithm is used, which is a particular realization of thecoordinate descent method (Luenberger, 1989). The filter and the image are consideredas dual variables. The algorithm alternately minimizes a cost function by descending withrespect to the filter or the image, while fixing the dual variable. In every alternating step, ˆa quadratic cost function L(f, h|f ) or L(f, h|ˆ) is minimized by the conjugate gradient gmethod. We note that this formulation is equivalent to minimization of a functional: 2 √ 2 √ 2 L(f , h) = ω(h ∗ f − g) L2 +λ1 ω 1 C1 ∗ f L2 +λ2 ω 2 C2 ∗ h L2 ,where f and h are image and blur kernel 2D real functions and · L2 is an L2 – norm. Regularization with another form of constraint has been considered in (Chan andWong, 1997), where the problem is formulated as a minimization of the functional: 2 L(f , h) = h ∗ f − g L2 +α1 | f |dx + α2 | h|dx. (6.3.10) Ω ΩThe proposed method is called total variation blind deconvolution (TV regularization).In Eq. (6.3.10) the regularization term has the form Ω | f |dx, called a total variation(TV) norm (Rudin et al., 1992). It follows the idea that the image consists of the smoothpatches, instead of being smooth everywhere, thus providing better recovering of imageedges.6.3.2 Image restoration in the frequency domainAll the restoration methods considered up to this point were derived in the space domain,though historically the first methods were designed in the frequency domain. Herein wesurvey briefly the most widely spread frequency domain restoration methods.
    • Chapter 6: Recognition of blurred images 110Wiener filterA fundamental result in filtering theory used commonly for image restoration is a Wienerfilter. Wiener filtering has been successfully used to filter images corrupted both by noiseand blurring. This filter gives the best estimate of the object from the observations inthe MSE sense. The Wiener filter frequency response is given as (Jain, 1989): H Sf f Sgg − Sηη HW = 2 S = . (6.3.11) H f f + Sηη HSggIn the case where only one observation is available, Sf f and Sgg are power spectrums ofideal and observed images, respectively, and Sηη is a power spectrum of the noise. Sincethe phase of the Wiener filter coincides with the phase of the inverse filter, it does notcompensate for phase distortions due to noise in the observations. In the absence of the blurring, the Wiener filter becomes: Sf f snr HW = = , (6.3.12) Sf f + Sηη snr + 1where snr = Sf f /Sηη is a signal-to-noise ratio. In practice, snr is defined as a ratiobetween variances of the blurred image and the noise (or 10 log10 snr, if signal-to-noiseratio is measured in Db) . This filter (6.3.12) is called the Wiener smoothing filter. It suppresses all frequencycomponents in which the signal-to-noise ratio is small and does not change the frequencycomponents when snr is large (snr 1). For images, Sf f is usually very small for highfrequencies, therefore the noise smoothing filter is a low pass filter. Another marginalcase is the absence of noise, in which the Wiener filter coincides with the inverse filterHW = H −1 . Since the blurring process is usually a low pass filter, the Wiener filter actsin this case as a high pass filter. In the presence of noise and blur, the Wiener filter achieves a compromise betweenlow-pass noise smoothing and high-pass inverse filtering, resulting in a band-limited filter.It is clear, nevertheless, that the Wiener filter is also unstable (like the inverse filter), ifthe frequency response is zero or close to it.Inverse and pseudo-inverse filtersAs has been already mentioned, in the case of the noise absence, the Wiener filter becomesan inverse one and requires stabilization. A standard stabilized version of the inverse filteris described by the following equation: 1 −1 H(w1 ,w2 ) if H(w1 , w2 ) ≥ 1 H (w1 , w2 ) = 0 otherwise
    • Chapter 6: Recognition of blurred images 111Instead we have used the next version of the pseudo-inverse filter in our simulations 1 H(w1 ,w2 ) if H(w1 , w2 ) ≥ 1 H −1 (w1 , w2 ) = 1 H(w1 ,w2 )+ 2 otherwiseThe choice of the 1 and 2 parameters defines the quality of the deblurred image. In oursimulations, they have been chosen by trial once for all the data set. It is known thatgreat care must be taken to obtain approximate solutions that achieve the proper balancebetween accuracy and stability. (Stark, 1987). Another nonlinear deblur filter is a rootfilter (see Section 6.2.1) that is also used for image enhancement.6.3.3 DenoisingDenoising may be considered a particular restoration method when the PSF of the bluroperator is a delta function. Thus, some of the methods described above are appropriatefor denoising (Rudin et al., 1992; You and Kaveh, 1996). We also consider two examplesof the rank algorithms (Yaroslavsky and Eden, 1996). Rank algorithms are especiallydesigned for noise reduction. They are based on the statistics extracted from the vari-ational row, that is a sequence of central pixel and its neighbors, ranked in increasingorder of their intensities. Different definitions of the neighborhood and variational rankstatistics lead to diverse rank algorithms. Rank statistics may be also obtained from localhistograms and are rather computationally efficient, when applied recursively. The mainadvantage of the rank algorithms is local adaptivity. Different denoising algorithms maybe also applied in the cascade. First, we consider an averaging technique, called peer group averaging (PGA), in whicha central pixel intensity is replaced by an average intensity of some predefined neighboringpixels, which are closest by intensity value. The number of pixels over which averaging isperformed is called the peer group size and it controls the amount of smoothing. The second method – the median filter, replaces the gray level intensity of each pixelby the median of its neighboring pixel intensities. This method is particularly effectivewhen the noise is spike-like. It is nonlinear, is very robust and preserves edge sharpness.6.4 ResultsOur experiments have shown that training with both schemes B and C (see Section 6.1.1)leads to recognition improvement compared with the training scheme A. We have alsoobserved that scheme B is superior to scheme C, but the difference between them is in-significant. Therefore, below we concentrate on ensembles obtained by using two training
    • Chapter 6: Recognition of blurred images 112schemes A and B, and postpone with summary comparison results for all three schemesuntil Section 6.4.6. All experiments are carried out on the TAU facial data-set.6.4.1 Image filteringIn the first group of experiments, the abilities of different ensembles to classify imagesprocessed by ideal and some typical low, band and high pass filters have been compared.Classification results are presented in Table 6.1 and some degraded images in Figure 6.3. Classification results for filtered data Types of Training scheme A Training scheme B corruption with extra blurred images Unconstrained Reconstruction Joined Unconstrained Reconstruction Joined λ=0 ensemble ensemble λ=0 ensemble ensemble ”Clean data” 12.8 12.8 13.5 9.5 10.8 8.8 Ideal low-pass 15.5 14.2 13.5 9.5 10.1 9.5 cutoff w = 10 Gaussian blur 19.6 16.2 16.9 14.2 11.5 10.8 with σ = 2 Out-of-focus blur with r = 5 20.9 20.9 17.6 16.2 10.8 11.5 Motion blur in the diagonal direction 32.4 26.4 26.4 29.7 24.3 19.6 with d = 5 Motion blur in the horizontal direction 21.6 24.3 19.6 16.9 14.9 12.2 with d = 5 Ideal band-pass 41.2 41.2 35.8 39.2 31.8 28.4 3 < w ≤ 10 DOG filter with σ1 = 1 and σ2 = 2 31.8 26.4 23.6 23.0 26.4 20.9 Ideal high pass 39.2 35.1 32.4 33.8 31.8 27.7 w >3 Root filter with 16.9 17.6 12.8 10.1 10.8 8.1 α = 0.6 Root filter with 12.8 13.5 12.8 9.5 9.5 8.1 α = 0.8 Table 6.1: Percent classification error for filtered data (TAU data set)Low-pass filteringWe have considered the ideal low-pass filter with cutoff w = 10 , the Gaussian blur withstandard deviation σ = 2, motion blur in diagonal and horizontal directions and theout-of-focus blur, all with blur propagation on 5 pixels. We note that for each of training schemes A–B, the unconstrained (λ = 0) ensembleis inferior to the reconstruction and joined ensembles in the blurred image recognition.In turn, the reconstruction ensembles are superior to the unconstrained ensembles. Forexample, for Gaussian blurred images the unconstrained ensemble of the training schemeA yields the misclassification rate of 19.6%, while the reconstruction ensemble produces
    • Chapter 6: Recognition of blurred images 11316.2%. For ensembles trained with the training scheme B, the misclassification rate fallsfrom 14.2% for the unconstrained ensemble to 11.5% for the reconstruction ensemble. Merging of the unconstrained and reconstruction ensembles improves classification re-sults further on. For example, for out-of-focus images, the joined ensemble of the trainingscheme A has the misclassification rate of 17.6%, while the reconstruction ensemble pro-duces 20.9%. For diagonal motion the joined ensemble of the training scheme B has themisclassification rate of 19.6% compared with 24.3% for the reconstruction ensemble. We note that reconstruction ensembles often give better classification results thanunconstrained ensembles and joined ensembles improve classification further on.Band-pass filteringBand-pass filtering is presented by the DOG filter with the size of on and off receptivefields equal to 1 and 2 pixels respectively, and ideal band-pass filtering with inner andouter cutoff radiuses equal to 3 and 10 respectively. Our experiments show that joinedensembles are better than reconstruction ensembles, which in most of the cases are betterthan unconstrained (λ = 0) ensembles. Therefore, for the training scheme A with testingon DOG filtered images, the misclassification rate falls from 31.8% for the unconstrainedensemble, to 26.4% for the reconstruction ensemble, and then to 23.6% for the joinedensemble. For the training scheme B the reconstruction ensemble is inferior to the un-constrained ensemble, but the joined ensemble is superior. Its classification performanceis 2.1% more than for the unconstrained λ = 0 ensemble. Finally, the joined ensem-ble with the scheme B improves the results by 10.9%, in comparison with the classicalunconstrained ensemble of the training scheme A.High-pass filteringHigh pass filtering is presented by the ideal high pass filter wih cutoff w = 3 and by theroot filter. Though images degraded with the high pass filter bear a resemblance to originalimages (Figure 6.3i), they are difficult for recognition. The smallest misclassification rateon this data is achieved by the joined ensemble of the training scheme B (27.7%). Whendegradation becomes less, recognition improves and even may be useful. Classificationresults on root filtered images are slightly better than the results for “clean” images.Surprisingly, humans also recognize slightly enhanced images better than the originalimages. Remarkably, joined ensembles are best in recognition of differently degradedimages.
    • Chapter 6: Recognition of blurred images 1146.4.2 Classification of noisy dataIn the following section, we shall test the performance of our scheme under realistic noiseand blur degradations. We first test the performance under various noise operations onnon-blurred objects in order to have a base line for comparison with the blurred results.Results of an ensemble of networks on noisy and restored images are presented in Table 6.2.Two kinds of noise, “Salt and Pepper” and Gaussian noise of small and large levels areconsidered. “Salt and Pepper” noise is implied with density parameters d = 0.2 andd = 0.6. Gaussian noise corresponds to snr = 10 and snr = 1. Median filter with awindow size 3 × 3 is used to denoise images corrupted with “Salt and Pepper” noise. Todenoise images degraded with Gaussian noise, peer group averaging (PGA) has been used.PGA window size 3 × 3 and group size ng = 5 have been chosen for snr = 10 and ng = 6for snr = 1. Noise and Restoration Types of Training scheme A Training scheme B corruption with extra blurred images Unconstrained Reconstruction Joined Unconstrained Reconstruction Joined λ=0 ensemble ensemble λ=0 ensemble ensemble ”Clean data” 12.8 12.8 13.5 9.5 10.8 8.8 “Salt and Pepper” noise with d = 0.2 25.0 20.3 20.3 20.3 18.2 14.2 Median filter denoising 13.5 12.8 12.8 9.5 8.8 8.8 “Salt and Pepper” noise with d = 0.6 70.3 66.9 69.6 81.8 76.4 74.3 Median filter denoising 25.0 20.3 21.6 20.9 20.9 14.9 Gaussian noise with snr = 10 13.5 13.5 12.8 8.1 10.8 8.1 PGA denoising with ng = 5 13.5 14.9 13.5 8.8 10.8 8.8 Gaussian noise with snr = 1 15.5 16.9 12.8 10.1 10.8 8.1 PGA denoising with ng = 6 14.9 15.5 12.8 10.8 12.8 8.8 Table 6.2: Percent classification error for noisy data (TAU data set) Examples of noisy and restored images are presented in Figure 6.4. We note thatclassification is more sensitive to “Salt and Pepper” noise than to Gaussian noise, whichmay be explained by the quasi-linear type of MLP network transformations. For a “Salt and Pepper” noise of density d = 0.6, 60% of the image pixels intensitiesare replaced by marginal intensity values, which leads to a very high misclassificationrate. Additional preprocessing by median filter significantly improves classification andgives the mild misclassification rate of 14.9% for the best joined ensemble of the trainingscheme B. Sensitivity of the network ensembles to Gaussian noise is small. Moreover, the joined
    • Chapter 6: Recognition of blurred images 115 Noisy Images a b c dFigure 6.4: a) An image contaminated with “Salt and Pepper noise” at 20% corruption.b) Results of the median smoothing in a window of size 3 × 3. c) An image contaminatedwith Gaussian noise with snr = 1. d) Results of the peer group averaging in a window ofsize 3 × 3 and with a peer group of size ng = 6.ensembles of both schemes A and B are insensitive to Gaussian noise and denoising, whichis carried out beforehand, even slightly spoils classification results.6.4.3 Gaussian blurThe classification results for Gaussian blurred images without noise interference and fortheir restored images are presented in Table 6.3. The Gaussian operator has the standarddeviation equal to σ = 2. Gaussian Blur and Restoration Types of Training scheme A Training scheme B corruption with extra blurred images Unconstrained Reconstruction Joined Unconstrained Reconstruction Joined λ=0 ensemble ensemble λ=0 ensemble ensemble ”Clean data” 12.8 12.8 13.5 9.5 10.8 8.8 Gaussian blur 19.6 16.2 16.9 14.2 11.5 10.8 with σ = 2 Pseudoinverse filter: with σ = 1.5 : 15.5 13.5 14.2 8.1 10.1 8.8 with σ = 2.0 : 13.5 13.5 12.8 9.5 10.8 8.8 with σ = 2.5: 15.5 15.5 12.8 9.5 10.1 7.4 Root filter: α = 0.6: 12.8 13.5 12.8 14.9 12.8 10.8 α = 0.8: 13.5 14.2 14.2 12.2 12.2 8.8 Table 6.3: Percent classification error for deblurred data The most sensitive to the Gaussian blur is the unconstrained λ = 0 ensemble of thetraining scheme A and the best is the joined ensemble of the training scheme B. For deblurring, pseudo-inverse and root filters have been used. In pseudo-inversefilter, the standard deviation of the Gaussian kernel is assumed to be known only approx-imately. The inverse Gaussian operator with an approximated standard deviation σ in ˆ
    • Chapter 6: Recognition of blurred images 116the frequency domain is given by: −1 Hσ (w) = exp(−2π 2 σ 2 w2 ). ˆ ˆ (6.4.13)Thus two main cases exist. In the first case, the guessed value is less than the originalσ < σ and image remains partially blurred with Gaussian filter. In the second case, theˆguessed value exceeds the original (ˆ > σ), which corresponds to filtering with high-pass σfilter that is given in the frequency domain by: √ Hβ (w) = exp(2π 2 β 2 w2 ), β= σ2 − σ2. ˆ (6.4.14)This analysis does not consider computational problems connected with the asymptotic −1behavior of Hσ (w) as w tends to infinity. Classification results with pseudo-inverse filtered images are presented in Table 6.3 inthe rows marked with “Pseudo-inverse filter” and restored images are given in Figure 6.5(d-f). Pseudo-inverse filter has been applied three times with approximated standarddeviations σ = 1.5, 2, 2.5. As expected, deblurring improves the classification results and ˆ Gaussian blur and restoration a b c d e fFigure 6.5: a) Image blurred with Gaussian filter with standard deviation σ = 2 b)Enhancement with root filter with α = 0.8 c) Enhancement with root filter with α = 0.6d) Pseudo-inverse filter with guessed σ = 1.5 e) Pseudo-inverse filter with guessed σ = 2f) Pseudo-inverse filter with guessed σ = 2.5the best one are for the joined ensemble trained with the scheme B. We note that both
    • Chapter 6: Recognition of blurred images 117joined ensembles classify pseudo-inverse deblurred images with σ = 2.5 slightly better ˆthan ”clean” data. We have observed a similar behavior for high-pass filtered data. Asimple enhancement with root filter also improves the classification results.6.4.4 Motion blur Motion Blur and Restoration Types of Training scheme A Training scheme B corruption with extra blurred images Unconstrained Reconstruction Joined Unconstrained Reconstruction Joined λ=0 ensemble ensemble λ=0 ensemble ensemble ”Clean data” 12.8 12.8 13.5 9.5 10.8 8.8 d = 5 pixels 21.6 24.3 19.6 16.9 14.9 12.2 snr=inf (no noise) deblurring 12.8 12.8 12.8 9.5 10.8 8.8 d = 5 pixels and Gaussian noise 20.9 24.3 19.6 16.2 15.5 12.2 snr=100 deblurring 13.5 14.2 13.5 9.5 10.8 8.8 d = 5 pixels and Gaussian noise 21.6 23.6 19.6 16.9 15.5 12.2 snr=10 smoothing and deblurring 14.9 14.9 14.2 9.5 10.8 9.5 d = 7 pixels 27.0 29.1 23.6 20.3 23.0 16.2 snr=inf (no noise) blind deconvolution 13.5 15.5 12.8 10.8 11.5 9.5Table 6.4: Percent misclassification rate for motion blurred and restored images. Motiontakes place in the horizontal direction and Gaussian noise is added. Motion propagationis given as a parameter d. Noise level is indicated as a signal-to-noise ratio snr, if noiseis absent snr = inf . MSE minimization with adaptive Tikhonov regularization is usedfor restoration. Lines marked with “deblurring” stand for deblurring with a known bluroperator. Table 6.4 presents classification results for images degraded as a result of horizontalmotion and additive Gaussian noise (Figure 6.6). As expected, with increase of the blurpropagation, classification declines. As we have already seen, the influence of noise isless dramatical, in particular, for joined ensembles. Indeed, negative role of the noise isrevealed during image restoration. The blur propagation may be estimated from the well-defined periodic structure of zero-crossing line locations of motion filter in the frequencydomain. However, this method is highly sensitive to noise. For restoration, MSE minimization with Tikhonov adaptive regularization is used. Inall experiments with motion propagation on d = 5 pixels, a motion filter is assumed to beknown. For noise degradation with snr = 10 (10 Db), a simple smoothing (averaging) inthe window of size 3 × 3 pixels is carried out before restoration. Classification after de-blurring of images degraded with small noise snr = 100 (20 Db) is the same as for “clean”
    • Chapter 6: Recognition of blurred images 118 Motion blur and deblur a b c dFigure 6.6: a) Motion blur with propagation on 5 pixels and Gaussian noise with snr = 10.b) Motion deblur using the constrained regularization method with the known blur filterand with the simple averaging in the window 3 × 3 before its application. c) Motion blurwith blur propagation equal to 7 pixels. d) Blindly restored image.images for both joined ensembles. For larger noise with snr = 10 (10 Db) classification isslightly worse. To restore the images blurred as a result of motion with blur propagation parameterd = 7 pixels in the absence of noise, the Tikhonov regularization for both image andfilter is applied. Since the direction of motion blur can be easier estimated than themotion propagation parameter, it is assumed to be known. The kernel support of theblur filter is taken to be 9 pixels in the motion direction. Initial guesses are the observedblurred image for an image and a delta function for a blurring operator. The resultsof this experiment are presented in the two last rows of Table 6.4. Though deblurredimages differ slightly visually from the “clean” data, their classification is the same as for“clean” data. The joined ensemble obtained using the training scheme B is the best inclassification of motion blurred and restored images. The classification results for images,blurred with Gaussian filter, and contaminated with Gaussian noise, along with theirdeblur using blind deconvolution are presented below.6.4.5 Blind deconvolutionThis section presents classification results for blindly deconvolved images. The blurredimages are obtained as a spatial convolution of the original images with Gaussian kernelwith standard deviation equal to σ = 2 and pruned to have a support 7 × 7 pixels.Blind deconvolution is done using the regularization approach to image identification andrestoration (You and Kaveh, 1996). The filter and image are assumed to be positiveand a kernel support is taken to be 15 × 15 pixels. The sum of filter kernel coefficientsand summary image intensity are normalized to 1. The initial guess for an image is
    • Chapter 6: Recognition of blurred images 119the degraded face and we start from a delta function filter, no symmetry constraints(Chan and Wong, 1997) are used. The regularization parameters are set by hand fromvisual appearance once and for all images. An image blurred, with a truncated Gaussianfilter, and contaminated with Gaussian noise of snr = 100, and its blind deconvolutionare presented in Figure 6.7. Classification results for two cases, with and without noise Blind deconvolution a b c dFigure 6.7: a) Image blurred with Gaussian filter with standard deviation σ = 2, prunedto a support area 7 × 7 and Gaussian noise with snr = 100. b) Blind deblurring of thedegraded image. c) Original filter. d) Found filter, pruned to the same support as theoriginal filter.interference, are presented in Table 6.5. Blind Deconvolution Types of Training scheme A Training scheme B corruption with extra blurred images Unconstrained Reconstruction Joined Unconstrained Reconstruction Joined λ=0 ensemble ensemble λ=0 ensemble ensemble ”Clean data” 12.8 12.8 13.5 9.5 10.8 8.8 Blur with pruned 18.2 16.2 17.6 10.1 10.1 10.8 Gaussian filter Blind deconvolution 12.8 13.5 12.8 9.5 10.8 8.1 Blur with pruned Gaussian filter and Gaussian 18.9 16.2 18.2 10.8 10.1 10.8 noise, snr = 100 Blind deconvolution 12.8 14.2 12.8 10.1 10.1 8.8Table 6.5: Percent misclassification rate for blurred and blindly deblurred images. Theimages are blurred with pruned Gaussian filter. We note that between ensembles obtained with the training scheme A, reconstructionensemble is the less sensitive to blurring and noise. Ensembles obtained with the trainingscheme B are less sensitive to noise and blur. The joined ensemble obtained with thetraining scheme B has the best classification performance.
    • Chapter 6: Recognition of blurred images 1206.4.6 All training schemes Recognition of blurred images via schemes A–C 30 Misclassification rate % 20 10 0 f e d c A b Ad C a Cd B BdFigure 6.8: Percent classification error bar graph for reconstructed images. Regressionensembles A-C correspond to joined ensembles obtained using recognition schemes A-C,respectively. Heights of the bars marked with Ad-Cd show misclassification of ensemblesA-C respectively on restored images. See also corresponding Table 6.6 for description ofdegradation types a-f. Summary classification results for joined ensembles, corresponding to all trainingschemes A–C, are presented in Figure 6.8 and Table 6.6. First, we observe that en-sembles of networks trained using the expanded training data-set are superior to thejoined ensemble trained without blurred images. Secondly, we note that both ensemblesB-C have about the same classification performance and ensemble B is slightly better.This may be explained by the drastic compression rate, that causes reconstructed imagesto look blurred in both cases and results in the similarity of two types of reconstructionconstraints (see Figure 6.9). The third and important observation is that training with blurred images seems to bemore important than restoration preprocessing. Indeed, recognition of restored imagesusing scheme A (column and bars marked with Ad) is inferior to degraded image recogni-tion using schemes B–C. However, as was already marked, usage of image preprocessing
    • Chapter 6: Recognition of blurred images 121 Blurred image recognition via joined ensembles Image degradation types Joined ensembles and deblur type Bd B Cd C Ad A a) clean and root filter α = 0.8 8.1 8.8 9.5 8.1 12.8 13.5 b) Gaussian blur σ = 2 root filter α = 0.8 8.8 10.8 8.8 10.8 14.2 16.9 c) Truncated Gaussian blur σ = 2 and Gaussian noise snr=20 Db 8.8 10.8 8.1 10.1 12.8 18.2 and blind deconvolution d) Motion blur d = 7 and blind deconvolution 9.5 16.2 10.8 16.9 12.8 23.6 e) Out-of-focus blur with a = 5 * 11.5 * 12.8 * 17.6 f ) DoG filter σ1 = 1 and σ2 = 2 * 20.9 * 20.9 * 23.6Table 6.6: Percent classification error for reconstructed images. Regression ensembles A-C correspond to joined ensembles obtained using recognition schemes A-C, respectively.Columns marked with Ad-Cd show misclassification of ensembles A-C respectively onrestored images. Information where restoration process was not done is marked with *.Experiments with TAU data-setbefore recognition schemes leads to improved classification results.6.5 ConclusionsTwo ways to improve the challenging problem of blurred image recognition were proposed:(i) Preprocess the blurred images using blind deconvolution methods before recognition;(ii) Apply our regularized reconstruction constraints technique (Chapter 3) to a trainingset that has been expanded by blurred images of some form. This forces the reconstructionoperator that is estimated during training to become less sensitive to the blur operation.For this reason, training without reconstruction using the expanded training set does notimprove results. Two training schemes with and without blurred images have been compared and dif-ferent network ensembles have been considered. The best classification scheme is thescheme that includes both the hybrid recognition/reconstruction architecture and usageof blurred images. The best network ensemble is the joined ensemble, obtained by mergingof the unconstrained and the reconstruction ensembles trained with blurred images.
    • Chapter 6: Recognition of blurred images 122 We have shown that the combination of both ways, the restoration and regularizedclassification approach are superior to each one separately. Since restoration techniquesare very sensitive to noise and require a priori knowledge or visual human interaction, it isimportant that the hybrid classification/reconstruction is less sensitive to the restorationparameters.
    • Chapter 6: Recognition of blurred images 123 Reconstruction of Gaussian blurred images Training scheme B Training scheme CFigure 6.9: Reconstruction of Gaussian blurred images by Neural Networks obtained usingtraining schemes B–C. Images in the top row from left to right are an original image, its“caricature” image and Gaussian blurred image. In the middle row, images reconstructedby Neural Networks with λ = 0.05, 0.15, 0.25 and with reconstruction defined by thetraining scheme B are presented. In the bottom row, images reconstructed by NeuralNetworks with λ = 0.05, 0.15, 0.25 and with reconstruction defined by the training schemeC are presented. Note that though images in the middle row are sharper than images inthe bottom row, they nevertheless look blurred.
    • Chapter 7Summary and future workIn this final chapter, we summarize the main contribution of the thesis and present severalpossible directions for future work.7.1 SummaryOur primary goal in this thesis was to improve the performance of a high dimensionalimage recognition task, by extracting a good hidden representation of the image data.We developed several approaches to achieve a good generalization in image recognition. First we developed a novel hybrid feed-forward reconstruction/recognition networkarchitecture, with two output sublayers for reconstruction and recognition, and one com-mon hidden layer shared by both tasks (Chapter 3). The network was trained to minimizeconcurrently MSE of reconstruction and recognition output sublayers. Though, a similar architecture was used previously (see Section 3.1.5), we first usedit for improving image recognition and gave a new interpretation of the hybrid networkas a tool to control bias via imposing a novel type of reconstruction bias constraints. Inaddition, we introduced a trade-off parameter λ that defines the influence of each of thetasks and is unknown a-priori. We have considered networks with different values of λ,instead of considering only a single value, as has been proposed previously. In addition, the network and its learning rule were interpreted in the MDL andBayesian frameworks. In Bayesian formulation, the network is trained to maximize theconditional joint probability of the reconstructed image and its class label given the ob-served image. In the proposed architecture, the reconstructed image and its class labelare independent given the observed image and under the assumption of a Gaussian distri-bution of the errors, this maximization leads to the proposed learning rule. The trade-offparameter λ emerges as a hyper-parameter and according to the Bayesian theory, theright approach is to integrate predictors over this parameter. If the initial weights of the 124
    • Chapter 7: Summary and future work 125feed-forward network are also considered as hyper-parameters, then the predictor f isgiven by: f (x) = fλ,w0 (x)p(λ, w0 |X )dw0 dλ (7.1.1) This interpretation has led us to the second approach to improve image recognition.We have proposed to replace the integration in Eq. 7.1.1 by a rough approximation viaensemble network averaging. Networks with a good recognition performance were includedin the ensemble and their posterior probabilities p(λ, w0 |X ) were assumed to be equal. It is well known, that ensemble averaging can reduce the variance portion of theprediction error. We have considered three ensemble types (Chapter 5): • Unconstrained ensemble, which corresponds to integration over w0 for λ = 0 • Reconstruction ensemble, which corresponds to integration over λ for fixed w0 • Joined ensemble, which is a combination of unconstrained and reconstruction en- sembles and corresponds to integration over both parametersWe have shown that the joined ensemble is superior to the reconstruction ensemble, whichin turn is superior to the unconstrained ensemble, in recognition of images degraded byGaussian and pulse noises as well as by partial occlusion or image blur. Our third contribution concerns especially in improving recognition of blurred images.It is well known, that in many practical recognition tasks, images appear blurred due tomotion, bad weather conditions and defocusing of cameras. Three ways were proposedfor improving blurred image recognition: 1. Expanding the training set with Gaussian blurred images 2. During training, constraining reconstruction of the blurred images to the original clean images 3. Application of state of the art restoration methods to the blurred images before using the hybrid architectureThe first two ways have led to two additional joined ensembles that we trained with extraGaussian images and reconstruction constraints. Reconstruction was either to the blurredimage or the clean (non-blurred) image (Chapter 6). We have shown that ensembles thatwere trained on extra (blurred) images had improved recognition performance on differentimage degradation types. In addition, we have shown that training with extra images
    • Chapter 7: Summary and future work 126combined with restoration techniques achieved robust and best recognition performanceunder a wide range of blur operators and parameters. Additional contribution of the thesis is developing hybrid networks with unsupervisedlearning constraints (Chapter 5), which were mainly used for comparison with reconstruc-tion constraints. We have shown that these constraints can also be used for improvingthe recognition performance instead, or in parallel with reconstruction constraints. In addition, we addressed the issue of a network interpretability by investigating thenetwork hidden representation and hidden weights (Appendix 5.7), and by the saliencymap construction (Section 5.5). In contrast, to explicit understanding what informationis encoded in the hidden space, the saliency map allows one to decide which features in theinput are more important. We showed that usage of the saliency maps further improvesrecognition of images degraded with “Salt and Pepper” noise.7.2 Directions for future workNon face data sets We have tested the proposed hybrid system on facial data sets.Faces, however, are a special type of stimuli where all pixels are important (Biederman andKalocsai, 1997). It should be interesting to test the hybrid architecture performance ondata sets of similar objects, such as military images (different kinds of tanks, ships, cars,etc.), medical images (different kinds of tumor cells) and astronomical images (images ofdifferent stars and galaxies).Ensemble interpretation In Appendix 5.7, hidden representations of single NeuralNetworks with reconstruction constraints were investigated. In addition it was noted,that network ensemble hidden representation is not well defined. However, another formof interpretation using the mean derivative (over networks and images) with respect tothe inputs for each of the classes (Intrator and Intrator, 1993) may be very interesting.Recurrent network architecture Images reconstructed by Neural Networks (whichwe called prototypes, see Section 5.5) are reduced representation of the original images,since a drastic compression occurs via the bottleneck architecture (see Figure 5.5). How-ever, as can be seen, prototypes corresponding to the same class look similar, whileprototypes corresponding to different classes look different. It is also clear that a goodreconstruction/recognition network has to be able to recognize its own prototype images.Table 7.1 presents recognition performance of the unconstrained and reconstruction en-sembles (see Chapter 3), when they are tested on the prototype images. These results
    • Chapter 7: Summary and future work 127 Classification error for reconstructed images Types of degradation Regression Ensembles Unconstrained Reconstruction A Reconstruction B ”Clean data” input 0.5 1.5 1 prototype 1.5 2.1 2.6 “Salt and Pepper” noise with d = 0.2 input 7.2 12.8 13.3 prototype 7.2 11.8 12.8 “nose” occlusion input 0.5 1.5 1.5 prototype 1.5 2.1 2.1 “half face” occlusion input 5.6 5.6 6.2 prototype 6.2 6.7 7.2 ”DOG 1-2” input 7.7 4.1 8.2 prototype 8.2 4.1 8.2 Table 7.1: Errors are given in percent (Pentland data-set).show that networks are better in recognition of the original images than their own pro-totypes. This can be corrected by propagating reconstructed images back to the inputlayer during learning. In other words, during learning we propose to extend the trainingset with extra images xe , which are a linear combination of the input x and its prototypeimage xp : xe = ρ(t)x + (1 − ρ(t))xp , ρ ∈ [0, 1],where ρ(t) is a non increasing function of the training epoch number t, equal to 1 at thebeginning and 0 at the infinity. This procedure may give better results that should betested by simulation.Network ensembles We considered ensembles corresponding to the simplest versionof integration (7.1.1) with equal posterior probability p(λ, w0 |X ). Though it is impossibleto find posterior probability p(λ, w0 |X ) analytically, it may be heuristically postulated.Therefore, integration (7.1.1) may be replaced by the weighted network ensemble aver-aging. We tried to use weights based on different error types between input and outputreconstruction layers, such as Euclidean metric or correlation measure and their soft ver-sion using the exp(−x) function. However, our preliminary experiments do not showsignificant recognition improvement. Since the hybrid networks solve both recognition and reconstruction tasks, it is reason-able to use the ensemble of hybrid networks for reconstruction. The obtained prototypemay be used for recognition by all the networks.
    • Chapter 7: Summary and future work 128Degradation invariance constraints We considered the simplest version of invari-ance constraints expanding the data with Gaussian blurred images. Another type ofinvariance constraints is the tangent prop constraint, that was used for a group of geo-metrical transformations (see Section 2.3.2). This type of constraints may be adapted fordifferent types of blur operations for both recognition and reconstruction tasks.Generalization It would be interesting to generalize the hybrid architecture in the di-rection taken by other generative models (Hinton and Ghahramani, 1997; Ullman, 1995).
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