This is a presentation of the paper on kernel analysis of deep networks.

1. Kernel Analysis of
Deep Networks
By:
Gregoire Montavon
Mikio L. Braun
Klaus-Robert Muller
(Technical University of Berlin)
JMLR 2011
Presented by:
Behrang mehrparvar
(University of Houston)
April 8th, 2014

2. Roadmap

Deep Learning

Goodness of Representations

Measuring goodness

Role of architecture

3. Deep Learning?

Distributed representation
− Less examples in regions
− Capture global structure

Depth
− Efficient representation

Abstraction
− Higher-level features

Flexibility
− Incorporate prior knowledge

4. Distributed Representation [1]

5. Depth [2]

6. Abstraction [?]

7. Problem Specification

Deep Learning is still a Black Box!

Theoretical aspect
− e.g. studying depth in sum-product networks

Analytical arguments
− e.g. analysis of depth

Experimental results
− e.g. performance in application domains

Visualization
− e.g. measuring invariance

8. Kernel Methods

Decouples learning algorithms from data representation

Kernel operator:
− Measures similarity between points
− All the prior knowledge of the learning problem

In this paper:
− Not a learning machine
− Abstraction tool to model the deep network

9. Kernel Methods (cont.)

Kernel Methods
− model the deep network

Used to quantify ...
− the goodness of representations
− the evolution of good representations

10. Hypothesis
1) Simpler and more accurate representation throughout the depth
2) Structure of the network (restrictions) define the speed of how
representations are formed
– Evolution from dist. of pixels to dist. of classes

11. Problem Specification

Problem: Role of depth in goodness of representation

Challenge: Definition and Measurement for goodness

Solution:
– Simplicity

Dimensionality: number of kernel PCs

Number of local variations
– Accuracy

Classification error

12. Hypothesis (Cont.)

13. Method
1) Train the deep network
2) Infer the representation of each layer
3) Apply kernel PCA on each layer representations
4) Project data points on first d eigenvectors
5) Analyze the results

14. Method (Analysis)

15. Why Kernels?
1) Incorporating prior knowledge
2) Measurable simplicity and accuracy
3) Theoretical framework and convergence bounds [3]
4) Flexibility

16. Dimensionality and Complexity

17. Dimensionality and Complexity (cont.)

18. Intuition

Accuracy
– Task-relevant information

Simplicity
– Number of allowed local variations in the inputs
space
– However, does not explain domain-specific
regularities
– Robust to number of samples
• vs. number of support vectors

19. Effects of Kernel mapping

20. Experiment setup

Datasets
– MNIST
– CIFAR

Tasks
– Supervised learning
– Transfer learning

Architectures
– Multilayer perceptron (MLP)
– Pretrained multilayer perceptron (PMLP)
– Convolutional neural networks (CNN)

21. Effect of Settings

22. Effect of Depth (Hyp. 1)

23. Observation

Higher layers
– More accurate representations
– More simple representations

24. Architectures

Multilayer Perceptrons
– No preconditioning on learning problem
– Prior: NONE

Pretrained Multilayer perceptrons
– Better represents the underlying representation
– Contains a certain part of soluton
– Prior: generative model of input

Convolutional Neural Networks
– Prior: Spatial invariance

25. Multilayer Perceptron [4]

26. Convolutional Neural Networks [4]

27. Effect of Architecture (Hyp. 2)

28. Observation

MNIST:
– MLP: Discriminating is solved greedily
– PMLP and CNN: postpone to last layers

CIFAR
– MLP: Doesn't discriminate till last layer
– PMLP and CNN: spread it to more layers
WHY?!
– Good observation, but no explanation!
– Hints: dataset, priors, etc. ?

29. Effect of Architecture (Cont.)

30. Observation

Regularities in PMLP and CNN
– Facilitate the construction of a structured
solution
– Controls the rate of discrimination at every level

31. Label Contribution of PCs

32. Comments

Strengths
– Important and interesting problem
– Simple and intuitive approach
– Well designed experiments
– Good analysis of results

Weaknesses
– Too many observations
• e.g. role of sigma in scale invariance
– explaining observations

33. Future works?

Experiments on Unsupervised Learning

Explaining the results

Analysis on biological neural systems?!

34. References
1) Bengio, Yoshua, and Olivier Delalleau. "On the expressive
power of deep architectures." Algorithmic Learning Theory.
Springer Berlin Heidelberg, 2011.
2) Poon, Hoifung, and Pedro Domingos. "Sum-product networks: A
new deep architecture." Computer Vision Workshops (ICCV
Workshops), 2011 IEEE International Conference on. IEEE, 2011.
3) Braun, Mikio L., Joachim M. Buhmann, and Klaus-Robert Müller. "On
relevant dimensions in kernel feature spaces." The Journal of
Machine Learning Research 9 (2008): 1875-1908.
4) http://deeplearning.net/

35. Thanks ...