The document compares two algorithms for hyperspectral image unmixing - one based on minimum volume constraint and one based on sum of squared distances constraint. It analyzes the performance of the two algorithms under different conditions like flatness of the endmember simplex, effects of initialization, and robustness to noise. The analysis shows that the sum of squared distances constraint performs better than the volume constraint for non-regular simplex shapes and is more robust to random initialization and noise. The comparison provides guidance on which constraint is more suitable for specific hyperspectral unmixing tasks.
An Efficient Clustering Method for Aggregation on Data FragmentsIJMER
Clustering is an important step in the process of data analysis with applications to numerous fields. Clustering ensembles, has emerged as a powerful technique for combining different clustering results to obtain a quality cluster. Existing clustering aggregation algorithms are applied directly to large number of data points. The algorithms are inefficient if the number of data points is large. This project defines an efficient approach for clustering aggregation based on data fragments. In fragment-based approach, a data fragment is any subset of the data. To increase the efficiency of the proposed approach, the clustering aggregation can be performed directly on data fragments under comparison measure and normalized mutual information measures for clustering aggregation, enhanced clustering aggregation algorithms are described. To show the minimal computational complexity. (Agglomerative, Furthest, and Local Search); nevertheless, which increases the accuracy.
In this video from the 2015 HPC User Forum in Broomfield, Barry Bolding from Cray presents: HPC + D + A = HPDA?
"The flexible, multi-use Cray Urika-XA extreme analytics platform addresses perhaps the most critical obstacle in data analytics today — limitation. Analytics problems are getting more varied and complex but the available solution technologies have significant constraints. Traditional analytics appliances lock you into a single approach and building a custom solution in-house is so difficult and time consuming that the business value derived from analytics fails to materialize. In contrast, the Urika-XA platform is open, high performing and cost effective, serving a wide range of analytics tools with varying computing demands in a single environment. Pre-integrated with the Hadoop and Spark frameworks, the Urika-XA system combines the benefits of a turnkey analytics appliance with a flexible, open platform that you can modify for future analytics workloads. This single-platform consolidation of workloads reduces your analytics footprint and total cost of ownership."
Learn more: http://www.cray.com/products/analytics/urika-xa
Watch the video presentation: http://wp.me/p3RLEV-3yR
Sign up for our insideBIGDATA Newsletter: http://insidebigdata.com/newsletter
PERFORMANCE ASSESSMENT OF CHAOTIC SEQUENCE DERIVED FROM BIFURCATION DEPENDENT...IJCNCJournal
In CDMA system, m-sequence and Gold codes are often utilized for spreading-despreading and
scrambling-descrambling operations. In a previous work, a design framework was created for generating
large family of codes from logistic map, which have comparable autocorrelation and cross correlation to
m-sequence and Gold codes. The purpose of this work is to evaluate the performance of these chaotic
codes in a CDMA environment. In the bit error rate (BER) simulation, matched filter, decorrelator and
MMSE receiver have been utilized. The received signal was modelled for synchronous CDMA uplink for
simulation simplicity purpose. Additive White Gaussian Noise channel model was assumed for the
simulation.
A Weighted Duality based Formulation of MIMO SystemsIJERA Editor
This work is based on the modeling and analysis of multiple-input multiple-output (MIMO) system in downlink communication system. We take into account a recent work on the ratio of quadratic forms to formulate the weight matrices of quadratic norm in a duality structure. This enables us to achieve exact solutions for MIMO system operating under Rayleigh fading channels. We outline couple of scenarios dependent on the structure of eigenvalues to investigate the system behavior. The results obtained are validated by means of Monte Carlo simulations.
Macromodel of High Speed Interconnect using Vector Fitting Algorithmijsrd.com
At high frequency efficient macromodeling of high speed interconnects is all time challenging task. We have presented systematic methodologies to generate rational function approximations of high-speed interconnects using vector fitting technique for any type of termination conditions and construct efficient multiport model, which is easily and directly compatible with circuit simulators.
An Efficient Clustering Method for Aggregation on Data FragmentsIJMER
Clustering is an important step in the process of data analysis with applications to numerous fields. Clustering ensembles, has emerged as a powerful technique for combining different clustering results to obtain a quality cluster. Existing clustering aggregation algorithms are applied directly to large number of data points. The algorithms are inefficient if the number of data points is large. This project defines an efficient approach for clustering aggregation based on data fragments. In fragment-based approach, a data fragment is any subset of the data. To increase the efficiency of the proposed approach, the clustering aggregation can be performed directly on data fragments under comparison measure and normalized mutual information measures for clustering aggregation, enhanced clustering aggregation algorithms are described. To show the minimal computational complexity. (Agglomerative, Furthest, and Local Search); nevertheless, which increases the accuracy.
In this video from the 2015 HPC User Forum in Broomfield, Barry Bolding from Cray presents: HPC + D + A = HPDA?
"The flexible, multi-use Cray Urika-XA extreme analytics platform addresses perhaps the most critical obstacle in data analytics today — limitation. Analytics problems are getting more varied and complex but the available solution technologies have significant constraints. Traditional analytics appliances lock you into a single approach and building a custom solution in-house is so difficult and time consuming that the business value derived from analytics fails to materialize. In contrast, the Urika-XA platform is open, high performing and cost effective, serving a wide range of analytics tools with varying computing demands in a single environment. Pre-integrated with the Hadoop and Spark frameworks, the Urika-XA system combines the benefits of a turnkey analytics appliance with a flexible, open platform that you can modify for future analytics workloads. This single-platform consolidation of workloads reduces your analytics footprint and total cost of ownership."
Learn more: http://www.cray.com/products/analytics/urika-xa
Watch the video presentation: http://wp.me/p3RLEV-3yR
Sign up for our insideBIGDATA Newsletter: http://insidebigdata.com/newsletter
PERFORMANCE ASSESSMENT OF CHAOTIC SEQUENCE DERIVED FROM BIFURCATION DEPENDENT...IJCNCJournal
In CDMA system, m-sequence and Gold codes are often utilized for spreading-despreading and
scrambling-descrambling operations. In a previous work, a design framework was created for generating
large family of codes from logistic map, which have comparable autocorrelation and cross correlation to
m-sequence and Gold codes. The purpose of this work is to evaluate the performance of these chaotic
codes in a CDMA environment. In the bit error rate (BER) simulation, matched filter, decorrelator and
MMSE receiver have been utilized. The received signal was modelled for synchronous CDMA uplink for
simulation simplicity purpose. Additive White Gaussian Noise channel model was assumed for the
simulation.
A Weighted Duality based Formulation of MIMO SystemsIJERA Editor
This work is based on the modeling and analysis of multiple-input multiple-output (MIMO) system in downlink communication system. We take into account a recent work on the ratio of quadratic forms to formulate the weight matrices of quadratic norm in a duality structure. This enables us to achieve exact solutions for MIMO system operating under Rayleigh fading channels. We outline couple of scenarios dependent on the structure of eigenvalues to investigate the system behavior. The results obtained are validated by means of Monte Carlo simulations.
Macromodel of High Speed Interconnect using Vector Fitting Algorithmijsrd.com
At high frequency efficient macromodeling of high speed interconnects is all time challenging task. We have presented systematic methodologies to generate rational function approximations of high-speed interconnects using vector fitting technique for any type of termination conditions and construct efficient multiport model, which is easily and directly compatible with circuit simulators.
A New Approach to Linear Estimation Problem in Multiuser Massive MIMO SystemsRadita Apriana
A novel approach for solving linear estimation problem in multi-user massive MIMO systems is
proposed. In this approach, the difficulty of matrix inversion is attributed to the incomplete definition of the
dot product. The general definition of dot product implies that the columns of channel matrix are always
orthogonal whereas, in practice, they may be not. If the latter information can be incorporated into dot
product, then the unknowns can be directly computed from projections without inverting the channel
matrix. By doing so, the proposed method is able to achieve an exact solution with a 25% reduction in
computational complexity as compared to the QR method. Proposed method is stable, offers an extra
flexibility of computing any single unknown, and can be implemented in just twelve lines of code.
CONCURRENT TERNARY GALOIS-BASED COMPUTATION USING NANO-APEX MULTIPLEXING NIBS...VLSICS Design
Novel layout realizations for congestion-free three-dimensional lattice networks using the corresponding carbon-based field emission controlled switching is introduced in this article. The developed nano-based implementations are performed in three dimensions to perform the required concurrent computations for
which two-dimensional implementations are a special case. The introduced realizations for congestion-free concurrent computations utilize the field-emission controlled switching devices that were presented in the first and second parts of the article for the solution of synthesis congestion and by utilizing field-emission from carbon nanotubes and nanotips. Since the concept of symmetry indices has been related to regular logic design, a more general method called Iterative Symmetry Indices Decomposition that produces regular three-dimensional lattice networks via carbon field-emission multiplexing is presented, where one obtains multi-stage decompositions whenever volume-specific layout constraints have to be satisfied. The introduced congestion-free nano-based lattice computations form new and important paths in regular lattice realizations, where applications include low-power IC design for the control of autonomous robots and for signal processing implementations.
Cross-Talk Control with Analog In-Memory-Compute for Artificial Intelligence ...Bruce Morton
This brief paper reports on a study of the feasibility of 8-bit precision in analog, in-memory computation of Multiply-Accumulate (MAC) sums, as proposed for implementation of efficient AI/ML hardware. With conventional read-out means and plausible memory array parasitic assumptions, cross-talk renders the 8-bit goal unattainable. In contrast, with the addition of spatial filter circuitry, results show MAC sum read-out can be made insensitive to parasitic cross-talk.
Exact network reconstruction from consensus signals and one eigen valueIJCNCJournal
The basic inverse problem in spectral graph theory consists in determining the graph given its eigenvalue
spectrum. In this paper, we are interested in a network of technological agents whose graph is unknown,
communicating by means of a consensus protocol. Recently, the use of artificial noise added to consensus
signals has been proposed to reconstruct the unknown graph, although errors are possible. On the other
hand, some methodologies have been devised to estimate the eigenvalue spectrum, but noise could interfere
with the elaborations. We combine these two techniques in order to simplify calculations and avoid
topological reconstruction errors, using only one eigenvalue. Moreover, we use an high frequency noise to
reconstruct the network, thus it is easy to filter the control signals after the graph identification. Numerical
simulations of several topologies show an exact and robust reconstruction of the graphs.
Stochastic Computing Correlation Utilization in Convolutional Neural Network ...TELKOMNIKA JOURNAL
In recent years, many applications have been implemented in embedded systems and mobile Internet of Things (IoT) devices that typically have constrained resources, smaller power budget, and exhibit "smartness" or intelligence. To implement computation-intensive and resource-hungry Convolutional Neural Network (CNN) in this class of devices, many research groups have developed specialized parallel accelerators using Graphical Processing Units (GPU), Field-Programmable Gate Arrays (FPGA), or Application-Specific Integrated Circuits (ASIC). An alternative computing paradigm called Stochastic Computing (SC) can implement CNN with low hardware footprint and power consumption. To enable building more efficient SC CNN, this work incorporates the CNN basic functions in SC that exploit correlation, share Random Number Generators (RNG), and is more robust to rounding error. Experimental results show our proposed solution provides significant savings in hardware footprint and increased accuracy for the SC CNN basic functions circuits compared to previous work.
Learning to Extrapolate Knowledge: Transductive Few-shot Out-of-Graph Link Pr...MLAI2
Many practical graph problems, such as knowledge graph construction and drug-drug interaction prediction, require to handle multi-relational graphs. However, handling real-world multi-relational graphs with Graph Neural Networks (GNNs) is often challenging due to their evolving nature, as new entities (nodes) can emerge over time. Moreover, newly emerged entities often have few links, which makes the learning even more difficult. Motivated by this challenge, we introduce a realistic problem of few-shot out-of-graph link prediction, where we not only predict the links between the seen and unseen nodes as in a conventional out-of-knowledge link prediction task but also between the unseen nodes, with only few edges per node. We tackle this problem with a novel transductive meta-learning framework which we refer to as Graph Extrapolation Networks (GEN). GEN meta-learns both the node embedding network for inductive inference (seen-to-unseen) and the link prediction network for transductive inference (unseen-to-unseen). For transductive link prediction, we further propose a stochastic embedding layer to model uncertainty in the link prediction between unseen entities. We validate our model on multiple benchmark datasets for knowledge graph completion and drug-drug interaction prediction. The results show that our model significantly outperforms relevant baselines for out-of-graph link prediction tasks.
COMPARATIVE PERFORMANCE ANALYSIS OF RNSC AND MCL ALGORITHMS ON POWER-LAW DIST...acijjournal
Cluster analysis of graph related problems is an important issue now-a-day. Different types of graph
clustering techniques are appeared in the field but most of them are vulnerable in terms of effectiveness
and fragmentation of output in case of real-world applications in diverse systems. In this paper, we will
provide a comparative behavioural analysis of RNSC (Restricted Neighbourhood Search Clustering) and
MCL (Markov Clustering) algorithms on Power-Law Distribution graphs. RNSC is a graph clustering
technique using stochastic local search. RNSC algorithm tries to achieve optimal cost clustering by
assigning some cost functions to the set of clusterings of a graph. This algorithm was implemented by A.
D. King only for undirected and unweighted random graphs. Another popular graph clustering
algorithm MCL is based on stochastic flow simulation model for weighted graphs. There are plentiful
applications of power-law or scale-free graphs in nature and society. Scale-free topology is stochastic i.e.
nodes are connected in a random manner. Complex network topologies like World Wide Web, the web of
human sexual contacts, or the chemical network of a cell etc., are basically following power-law
distribution to represent different real-life systems. This paper uses real large-scale power-law
distribution graphs to conduct the performance analysis of RNSC behaviour compared with Markov
clustering (MCL) algorithm. Extensive experimental results on several synthetic and real power-law
distribution datasets reveal the effectiveness of our approach to comparative performance measure of
these algorithms on the basis of cost of clustering, cluster size, modularity index of clustering results and
normalized mutual information (NMI).
A simple framework for contrastive learning of visual representationsDevansh16
Link: https://machine-learning-made-simple.medium.com/learnings-from-simclr-a-framework-contrastive-learning-for-visual-representations-6c145a5d8e99
If you'd like to discuss something, text me on LinkedIn, IG, or Twitter. To support me, please use my referral link to Robinhood. It's completely free, and we both get a free stock. Not using it is literally losing out on free money.
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My Instagram: https://rb.gy/gmvuy9
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My Substack: https://devanshacc.substack.com/
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Get a free stock on Robinhood: https://join.robinhood.com/fnud75
This paper presents SimCLR: a simple framework for contrastive learning of visual representations. We simplify recently proposed contrastive self-supervised learning algorithms without requiring specialized architectures or a memory bank. In order to understand what enables the contrastive prediction tasks to learn useful representations, we systematically study the major components of our framework. We show that (1) composition of data augmentations plays a critical role in defining effective predictive tasks, (2) introducing a learnable nonlinear transformation between the representation and the contrastive loss substantially improves the quality of the learned representations, and (3) contrastive learning benefits from larger batch sizes and more training steps compared to supervised learning. By combining these findings, we are able to considerably outperform previous methods for self-supervised and semi-supervised learning on ImageNet. A linear classifier trained on self-supervised representations learned by SimCLR achieves 76.5% top-1 accuracy, which is a 7% relative improvement over previous state-of-the-art, matching the performance of a supervised ResNet-50. When fine-tuned on only 1% of the labels, we achieve 85.8% top-5 accuracy, outperforming AlexNet with 100X fewer labels.
Comments: ICML'2020. Code and pretrained models at this https URL
Subjects: Machine Learning (cs.LG); Computer Vision and Pattern Recognition (cs.CV); Machine Learning (stat.ML)
Cite as: arXiv:2002.05709 [cs.LG]
(or arXiv:2002.05709v3 [cs.LG] for this version)
Submission history
From: Ting Chen [view email]
[v1] Thu, 13 Feb 2020 18:50:45 UTC (5,093 KB)
[v2] Mon, 30 Mar 2020 15:32:51 UTC (5,047 KB)
[v3] Wed, 1 Jul 2020 00:09:08 UTC (5,829 KB)
A New Approach to Linear Estimation Problem in Multiuser Massive MIMO SystemsRadita Apriana
A novel approach for solving linear estimation problem in multi-user massive MIMO systems is
proposed. In this approach, the difficulty of matrix inversion is attributed to the incomplete definition of the
dot product. The general definition of dot product implies that the columns of channel matrix are always
orthogonal whereas, in practice, they may be not. If the latter information can be incorporated into dot
product, then the unknowns can be directly computed from projections without inverting the channel
matrix. By doing so, the proposed method is able to achieve an exact solution with a 25% reduction in
computational complexity as compared to the QR method. Proposed method is stable, offers an extra
flexibility of computing any single unknown, and can be implemented in just twelve lines of code.
CONCURRENT TERNARY GALOIS-BASED COMPUTATION USING NANO-APEX MULTIPLEXING NIBS...VLSICS Design
Novel layout realizations for congestion-free three-dimensional lattice networks using the corresponding carbon-based field emission controlled switching is introduced in this article. The developed nano-based implementations are performed in three dimensions to perform the required concurrent computations for
which two-dimensional implementations are a special case. The introduced realizations for congestion-free concurrent computations utilize the field-emission controlled switching devices that were presented in the first and second parts of the article for the solution of synthesis congestion and by utilizing field-emission from carbon nanotubes and nanotips. Since the concept of symmetry indices has been related to regular logic design, a more general method called Iterative Symmetry Indices Decomposition that produces regular three-dimensional lattice networks via carbon field-emission multiplexing is presented, where one obtains multi-stage decompositions whenever volume-specific layout constraints have to be satisfied. The introduced congestion-free nano-based lattice computations form new and important paths in regular lattice realizations, where applications include low-power IC design for the control of autonomous robots and for signal processing implementations.
Cross-Talk Control with Analog In-Memory-Compute for Artificial Intelligence ...Bruce Morton
This brief paper reports on a study of the feasibility of 8-bit precision in analog, in-memory computation of Multiply-Accumulate (MAC) sums, as proposed for implementation of efficient AI/ML hardware. With conventional read-out means and plausible memory array parasitic assumptions, cross-talk renders the 8-bit goal unattainable. In contrast, with the addition of spatial filter circuitry, results show MAC sum read-out can be made insensitive to parasitic cross-talk.
Exact network reconstruction from consensus signals and one eigen valueIJCNCJournal
The basic inverse problem in spectral graph theory consists in determining the graph given its eigenvalue
spectrum. In this paper, we are interested in a network of technological agents whose graph is unknown,
communicating by means of a consensus protocol. Recently, the use of artificial noise added to consensus
signals has been proposed to reconstruct the unknown graph, although errors are possible. On the other
hand, some methodologies have been devised to estimate the eigenvalue spectrum, but noise could interfere
with the elaborations. We combine these two techniques in order to simplify calculations and avoid
topological reconstruction errors, using only one eigenvalue. Moreover, we use an high frequency noise to
reconstruct the network, thus it is easy to filter the control signals after the graph identification. Numerical
simulations of several topologies show an exact and robust reconstruction of the graphs.
Stochastic Computing Correlation Utilization in Convolutional Neural Network ...TELKOMNIKA JOURNAL
In recent years, many applications have been implemented in embedded systems and mobile Internet of Things (IoT) devices that typically have constrained resources, smaller power budget, and exhibit "smartness" or intelligence. To implement computation-intensive and resource-hungry Convolutional Neural Network (CNN) in this class of devices, many research groups have developed specialized parallel accelerators using Graphical Processing Units (GPU), Field-Programmable Gate Arrays (FPGA), or Application-Specific Integrated Circuits (ASIC). An alternative computing paradigm called Stochastic Computing (SC) can implement CNN with low hardware footprint and power consumption. To enable building more efficient SC CNN, this work incorporates the CNN basic functions in SC that exploit correlation, share Random Number Generators (RNG), and is more robust to rounding error. Experimental results show our proposed solution provides significant savings in hardware footprint and increased accuracy for the SC CNN basic functions circuits compared to previous work.
Learning to Extrapolate Knowledge: Transductive Few-shot Out-of-Graph Link Pr...MLAI2
Many practical graph problems, such as knowledge graph construction and drug-drug interaction prediction, require to handle multi-relational graphs. However, handling real-world multi-relational graphs with Graph Neural Networks (GNNs) is often challenging due to their evolving nature, as new entities (nodes) can emerge over time. Moreover, newly emerged entities often have few links, which makes the learning even more difficult. Motivated by this challenge, we introduce a realistic problem of few-shot out-of-graph link prediction, where we not only predict the links between the seen and unseen nodes as in a conventional out-of-knowledge link prediction task but also between the unseen nodes, with only few edges per node. We tackle this problem with a novel transductive meta-learning framework which we refer to as Graph Extrapolation Networks (GEN). GEN meta-learns both the node embedding network for inductive inference (seen-to-unseen) and the link prediction network for transductive inference (unseen-to-unseen). For transductive link prediction, we further propose a stochastic embedding layer to model uncertainty in the link prediction between unseen entities. We validate our model on multiple benchmark datasets for knowledge graph completion and drug-drug interaction prediction. The results show that our model significantly outperforms relevant baselines for out-of-graph link prediction tasks.
COMPARATIVE PERFORMANCE ANALYSIS OF RNSC AND MCL ALGORITHMS ON POWER-LAW DIST...acijjournal
Cluster analysis of graph related problems is an important issue now-a-day. Different types of graph
clustering techniques are appeared in the field but most of them are vulnerable in terms of effectiveness
and fragmentation of output in case of real-world applications in diverse systems. In this paper, we will
provide a comparative behavioural analysis of RNSC (Restricted Neighbourhood Search Clustering) and
MCL (Markov Clustering) algorithms on Power-Law Distribution graphs. RNSC is a graph clustering
technique using stochastic local search. RNSC algorithm tries to achieve optimal cost clustering by
assigning some cost functions to the set of clusterings of a graph. This algorithm was implemented by A.
D. King only for undirected and unweighted random graphs. Another popular graph clustering
algorithm MCL is based on stochastic flow simulation model for weighted graphs. There are plentiful
applications of power-law or scale-free graphs in nature and society. Scale-free topology is stochastic i.e.
nodes are connected in a random manner. Complex network topologies like World Wide Web, the web of
human sexual contacts, or the chemical network of a cell etc., are basically following power-law
distribution to represent different real-life systems. This paper uses real large-scale power-law
distribution graphs to conduct the performance analysis of RNSC behaviour compared with Markov
clustering (MCL) algorithm. Extensive experimental results on several synthetic and real power-law
distribution datasets reveal the effectiveness of our approach to comparative performance measure of
these algorithms on the basis of cost of clustering, cluster size, modularity index of clustering results and
normalized mutual information (NMI).
A simple framework for contrastive learning of visual representationsDevansh16
Link: https://machine-learning-made-simple.medium.com/learnings-from-simclr-a-framework-contrastive-learning-for-visual-representations-6c145a5d8e99
If you'd like to discuss something, text me on LinkedIn, IG, or Twitter. To support me, please use my referral link to Robinhood. It's completely free, and we both get a free stock. Not using it is literally losing out on free money.
Check out my other articles on Medium. : https://rb.gy/zn1aiu
My YouTube: https://rb.gy/88iwdd
Reach out to me on LinkedIn. Let's connect: https://rb.gy/m5ok2y
My Instagram: https://rb.gy/gmvuy9
My Twitter: https://twitter.com/Machine01776819
My Substack: https://devanshacc.substack.com/
Live conversations at twitch here: https://rb.gy/zlhk9y
Get a free stock on Robinhood: https://join.robinhood.com/fnud75
This paper presents SimCLR: a simple framework for contrastive learning of visual representations. We simplify recently proposed contrastive self-supervised learning algorithms without requiring specialized architectures or a memory bank. In order to understand what enables the contrastive prediction tasks to learn useful representations, we systematically study the major components of our framework. We show that (1) composition of data augmentations plays a critical role in defining effective predictive tasks, (2) introducing a learnable nonlinear transformation between the representation and the contrastive loss substantially improves the quality of the learned representations, and (3) contrastive learning benefits from larger batch sizes and more training steps compared to supervised learning. By combining these findings, we are able to considerably outperform previous methods for self-supervised and semi-supervised learning on ImageNet. A linear classifier trained on self-supervised representations learned by SimCLR achieves 76.5% top-1 accuracy, which is a 7% relative improvement over previous state-of-the-art, matching the performance of a supervised ResNet-50. When fine-tuned on only 1% of the labels, we achieve 85.8% top-5 accuracy, outperforming AlexNet with 100X fewer labels.
Comments: ICML'2020. Code and pretrained models at this https URL
Subjects: Machine Learning (cs.LG); Computer Vision and Pattern Recognition (cs.CV); Machine Learning (stat.ML)
Cite as: arXiv:2002.05709 [cs.LG]
(or arXiv:2002.05709v3 [cs.LG] for this version)
Submission history
From: Ting Chen [view email]
[v1] Thu, 13 Feb 2020 18:50:45 UTC (5,093 KB)
[v2] Mon, 30 Mar 2020 15:32:51 UTC (5,047 KB)
[v3] Wed, 1 Jul 2020 00:09:08 UTC (5,829 KB)
Performance Analysis of Massive MIMO Downlink System with Imperfect Channel S...IJRES Journal
We investigate the ergodic sum rate and required transmit power of a single-cell massive
multiple-input multiple-output (MIMO) downlink system. The system considered in this paper is based on two
linear beamforming schemes, that is, maximum ratio transmission (MRT) beamforming and zero-forcing (ZF)
beamforming. What’s more, we use minimum mean square error (MMSE) channel estimation to get imperfect
channel state information (CSI). Compared with the perfect CSI case, both theoretical analysis and simulation
results show that the system performance is different when the imperfect CSI is taken into account.
Joint3DShapeMatching - a fast approach to 3D model matching using MatchALS 3...Mamoon Ismail Khalid
we extend the global optimization-based
approach of jointly matching a set of images to jointly
matching a set of 3D meshes. The estimated correspon
dences simultaneously maximize pairwise feature affini
ties and cycle consistency across multiple models. We
show that the low-rank matrix recovery problem can be
efficiently applied to the 3D meshes as well. The fast
alternating minimization algorithm helps to handle real
world practical problems with thousands of features. Ex
perimental results show that, unlike the state-of-the-art
algorithm which rely on semi-definite programming, our
algorithm provides an order of magnitude speed-up along
with competitive performance. Along with the joint shape
matching we propose an approach to apply a distortion
term in pairwise matching, which helps in successfully
matching the reflexive sub-parts of two models distinc
tively. In the end, we demonstrate the applicability of
the algorithm to match a set of 3D meshes of the SCAPE
benchmark database
Black-box modeling of nonlinear system using evolutionary neural NARX modelIJECEIAES
Nonlinear systems with uncertainty and disturbance are very difficult to model using mathematic approach. Therefore, a black-box modeling approach without any prior knowledge is necessary. There are some modeling approaches have been used to develop a black box model such as fuzzy logic, neural network, and evolution algorithms. In this paper, an evolutionary neural network by combining a neural network and a modified differential evolution algorithm is applied to model a nonlinear system. The feasibility and effectiveness of the proposed modeling are tested on a piezoelectric actuator SISO system and an experimental quadruple tank MIMO system.
COMPARISON OF WAVELET NETWORK AND LOGISTIC REGRESSION IN PREDICTING ENTERPRIS...ijcsit
Enterprise financial distress or failure includes bankruptcy prediction, financial distress, corporate performance prediction and credit risk estimation. The aim of this paper is that using wavelet networks innon-linear combination prediction to solve ARMA (Auto-Regressive and Moving Average) model problem.ARMA model need estimate the value of all parameters in the model, it has a large amount of computation.Under this aim, the paper provides an extensive review of Wavelet networks and Logistic regression. Itdiscussed the Wavelet neural network structure, Wavelet network model training algorithm, Accuracy rateand error rate (accuracy of classification, Type I error, and Type II error). The main research opportunity exist a proposed of business failure prediction model (wavelet network model and logistic regression
model). The empirical research which is comparison of Wavelet Network and Logistic Regression on training and forecasting sample, the result shows that this wavelet network model is high accurate and the overall prediction accuracy, Type Ⅰerror and Type Ⅱ error, wavelet networks model is better thanlogistic regression model.
LOGNORMAL ORDINARY KRIGING METAMODEL IN SIMULATION OPTIMIZATIONorajjournal
This paper presents a lognormal ordinary kriging (LOK) metamodel algorithm and its application to
optimize a stochastic simulation problem. Kriging models have been developed as an interpolation method
in geology. They have been successfully used for the deterministic simulation optimization (SO) problem. In
recent years, kriging metamodeling has attracted a growing interest with stochastic problems. SO
researchers have begun using ordinary kriging through global optimization in stochastic systems. The
goals of this study are to present LOK metamodel algorithm and to analyze the result of the application
step-by-step. The results show that LOK is a powerful alternative metamodel in simulation optimization
when the data are too skewed.
MODIFIED LLL ALGORITHM WITH SHIFTED START COLUMN FOR COMPLEXITY REDUCTIONijwmn
Multiple-input multiple-output (MIMO) systems are playing an important role in the recent wireless
communication. The complexity of the different systems models challenge different researches to get a good
complexity to performance balance. Lattices Reduction Techniques and Lenstra-Lenstra-Lovàsz (LLL)
algorithm bring more resources to investigate and can contribute to the complexity reduction purposes.
In this paper, we are looking to modify the LLL algorithm to reduce the computation operations by
exploiting the structure of the upper triangular matrix without “big” performance degradation. Basically,
the first columns of the upper triangular matrix contain many zeroes, so the algorithm will perform several
operations with very limited income. We are presenting a performance and complexity study and our
proposal show that we can gain in term of complexity while the performance results remains almost the
same.
The Positive Effects of Fuzzy C-Means Clustering on Supervised Learning Class...Waqas Tariq
Selection of inputs is one of the most substantial components of classification algorithms for data mining and pattern recognition problems since even the best classifier will perform badly if the inputs are not selected very well. Big data and computational complexity are main cause of bad performance and low accuracy for classical classifiers. In other words, the complexity of classifier method is inversely proportional with its classification efficiency. For this purpose, two hybrid classifiers have been developed by using both type-1 and type-2 fuzzy c-means clustering with cascaded a classifier. In this proposed classifier, a large number of data points are reduced by using fuzzy c-means clustering before applied to a classifier algorithm as inputs. The aim of this study is to investigate the effect of fuzzy clustering on well-known and useful classifiers such as artificial neural networks (ANN) and support vector machines (SVM). Then the role of positive effects of these proposed algorithms were investigated on applied different data sets.
The Positive Effects of Fuzzy C-Means Clustering on Supervised Learning Class...CSCJournals
Selection of inputs is one of the most substantial components of classification algorithms for data mining and pattern recognition problems since even the best classifier will perform badly if the inputs are not selected very well. Big data and computational complexity are main cause of bad performance and low accuracy for classical classifiers. In other words, the complexity of classifier method is inversely proportional with its classification efficiency. For this purpose, two hybrid classifiers have been developed by using both type-1 and type-2 fuzzy c-means clustering with cascaded a classifier. In this proposed classifier, a large number of data points are reduced by using fuzzy c-means clustering before applied to a classifier algorithm as inputs. The aim of this study is to investigate the effect of fuzzy clustering on well-known and useful classifiers such as artificial neural networks (ANN) and support vector machines (SVM). Then the role of positive effects of these proposed algorithms were investigated on applied different data sets.
Mitigating Interference to GPS Operation Using Variable Forgetting Factor Bas...IJCNCJournal
In this paper, an interference method based on signal processing is proposed. The approach is based on
utilizing the maximum likelihood properties of the received signal. The approach is built on maximizing the
probability of the desired data. The GPS data, which is constructed using Binary Phase Shift Keying
(BPSK) modulation, is transmitted as “1’s” and as “0’s.” carried on 1575.42MHz carrier called the L1
frequency. The statistics of the GPS data and interference are utilized in terms of their distribution and
variance. The statistics are used to update (adaptively) the forgetting factor (Lambda) of the Recursive
Least Squares (RLS) filter. The proposed method is called Maximum Likelihood Variable Forgetting Factor
(ML VFF). The adaptive update takes on assigning lambda to the maximum of the probabilities of the
symbols based on the statistics mentioned.
Parallel Batch-Dynamic Graphs: Algorithms and Lower BoundsSubhajit Sahu
Highlighted notes on Parallel Batch-Dynamic Graphs: Algorithms and Lower Bounds.
While doing research work under Prof. Kishore Kothapalli.
Laxman Dhulipala, David Durfee, Janardhan Kulkarni, Richard Peng, Saurabh Sawlani, Xiaorui Sun:
Parallel Batch-Dynamic Graphs: Algorithms and Lower Bounds. SODA 2020: 1300-1319
In this paper we study the problem of dynamically maintaining graph properties under batches of edge insertions and deletions in the massively parallel model of computation. In this setting, the graph is stored on a number of machines, each having space strongly sublinear with respect to the number of vertices, that is, n for some constant 0 < < 1. Our goal is to handle batches of updates and queries where the data for each batch fits onto one machine in constant rounds of parallel computation, as well as to reduce the total communication between the machines. This objective corresponds to the gradual buildup of databases over time, while the goal of obtaining constant rounds of communication for problems in the static setting has been elusive for problems as simple as undirected graph connectivity. We give an algorithm for dynamic graph connectivity in this setting with constant communication rounds and communication cost almost linear in terms of the batch size. Our techniques combine a new graph contraction technique, an independent random sample extractor from correlated samples, as well as distributed data structures supporting parallel updates and queries in batches. We also illustrate the power of dynamic algorithms in the MPC model by showing that the batched version of the adaptive connectivity problem is P-complete in the centralized setting, but sub-linear sized batches can be handled in a constant number of rounds. Due to the wide applicability of our approaches, we believe it represents a practically-motivated workaround to the current difficulties in designing more efficient massively parallel static graph algorithms.
Parallel Batch-Dynamic Graphs: Algorithms and Lower BoundsSubhajit Sahu
In this paper we study the problem of dynamically
maintaining graph properties under batches of edge
insertions and deletions in the massively parallel model
of computation. In this setting, the graph is stored
on a number of machines, each having space strongly
sublinear with respect to the number of vertices, that
is, n
for some constant 0 < < 1. Our goal is to
handle batches of updates and queries where the data
for each batch fits onto one machine in constant rounds
of parallel computation, as well as to reduce the total
communication between the machines. This objective
corresponds to the gradual buildup of databases over
time, while the goal of obtaining constant rounds of
communication for problems in the static setting has
been elusive for problems as simple as undirected graph
connectivity.
We give an algorithm for dynamic graph connectivity
in this setting with constant communication rounds and
communication cost almost linear in terms of the batch
size. Our techniques combine a new graph contraction
technique, an independent random sample extractor from
correlated samples, as well as distributed data structures
supporting parallel updates and queries in batches.
We also illustrate the power of dynamic algorithms in
the MPC model by showing that the batched version
of the adaptive connectivity problem is P-complete in
the centralized setting, but sub-linear sized batches can
be handled in a constant number of rounds. Due to
the wide applicability of our approaches, we believe
it represents a practically-motivated workaround to the
current difficulties in designing more efficient massively
parallel static graph algorithms.
Similar to COMPARISON OF VOLUME AND DISTANCE CONSTRAINT ON HYPERSPECTRAL UNMIXING (20)
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https://arxiv.org/abs/2306.08302
2. Microsoft Research's GraphRAG paper and a review paper on various uses of knowledge graphs:
https://www.microsoft.com/en-us/research/blog/graphrag-unlocking-llm-discovery-on-narrative-private-data/
2. 14 Computer Science & Information Technology (CS & IT)
(MVC-NMF) [10], iterative constrained endmember method (ICE) [11], robust nonnegative
matrix factorization [19], minimum volume simplex analysis(MVSA) [20].
2. RELATED WORK
Since there is no requirement on pure signatures for the algorithms in the second group, they have
been widely used in hyperspectral unmixing [9] [10] [11]. Among them, NMF with minimum
volume constraint (VC) [10] considers simplex volume composed of the unknown endmembers
during endmember extraction. VC adopts the commonly used formula to measure the ‘volume’ of
simplex enclosed by endmembers. Though VC can efficiently restrain simplex volume, it often
involves massive computation. On the other hand, ICE [11] imposes sum of squared distances
constraint (SSD) on the original objective function, which can generally achieve satisfied result.
Though algorithms based on VC and SSD have been proposed for several years, there hasn’t been
any further analysis and comparison of these two methods. Thus, this paper mainly gives a
detailed comparison of these two methods under various situations. The comparison analysis aims
to give an instruction on how to choose the constraint under certain situation. To achieve this, the
comparison includes flatness of simplex analysis, initialization analysis and robustness to noise
analysis.
3. HYPERSPECTRAL UNMIXING ALGORITHM
In this section, we’ll briefly introduce linear mixing model and unmixing algorithms based on VC
and SSD.
3.1. Linear Mixture Model (LMM)
LMM [2], [12] assumes that the hyperspectral data is a linear combination of endmember spectra,
with the weights being proportions. Mathematically, the model is given as:
x As ε= + ,
subjected to:
1
0, 1, , 1
M
k k
k
s k M s
=
≥ = =∑L , , (1)
where, x is L -dimensional vector ( L is the number of bands) which is one of the pixel in image,
s denotes corresponding abundance, ε represents possible errors. In real data processing,
abundance should satisfy two constraints called as nonnegative constraint and sum-to-one
constraint, as shown in (1).
The matrix involving all pixels in image is shown as Equation(2)
X AS= +Θ , (2)
where 1 2[ , , ]X x x xN= L represents hyperspectral data which is assumed to be composed with
material signatures 1 2[ , , ]A a a aM= L and abundance fractions 1 2[ , , ]S s s s N= L . N is the
number of pixel. M is the number of endmembers. Θ is the error matrix.
3.2. Constraints on Endmembers
For real data, there are substantial local minimum problem due to non-convexity of unconstrained
objection function. As shown in
3. Computer Science & Information Technology (CS & IT) 15
Figure 1, the red polyline indicates real simplex for hyperspectral data. Meanwhile, blue and
black polyline is the solution obtained with different initial value. Since there must be
corresponding abundance if endmembers enclose all scatters, it is necessary to put certain
constraint on endmembers, as shown in (3):
21
( , ) || || ( )
2
A S X AS AFf Jλ= − + (3)
where, ( )AJ is the constraint added to endmember, λ is regularization factor to tradeoff the
reconstruction error and constraint.
Before minimizing (3), several preprocessing steps will be taken to remove noises and reduce the
dimension of original data, which aims to reduce computation complexity. Then, appropriate
optimal strategy is used to minimize (3) and update A and S iteratively: first, given endmember
matrix A , calculate the abundance matrix S by the optimal strategy. Then update A by fixed
S in the same way. After several iterations, (3) will approach its minimum value with A and S
well-decided.
Bandb
Band a
Figure 1. Endmember extraction by unconstraint algorithms. There exist many local minimum solutions
due to the non-convex property. Although many solutions have relatively minor linear square error, the
obtained points are still far from scattering, which cannot be regard as appropriate endmembers.
The volume and distance constraints are briefed as follows:
1. Minimum Volume Constraint
VC minimize volume [15] of simplex in its model. In VC, the expression of ( )AVJ is as for
those.
2 11
( ) det ( )
2( 1)!
A
M A
T
M
VJ
=
−
(4)
After adding volume constraint, volume of simplex will be compressed as small as possible.
Meanwhile, the hyperspectral data reconstructed by the extracted endmembers and corresponding
abundance matrix can also close to real data. Therefore, we can get relatively accurate solution
for endmember and abundance.
2. Sum of Squared Distance Constraint
In ICE [11], the constraint which minimizes SSD among several endmembers on hyperplane is
adopted and given as equation (5). Like VC, SSD can also efficiently control the shape of simplex
during the iteration by minimizing the distance between any two endmembers.
1
1 1
( ) ( ) ( )A a a a a
M M
T
D k l k l
k l k
J
−
= = +
= − −∑ ∑ , (5)
Where, M is the number of endmembers, ak and al are any two endmember vectors.
4. 16 Computer Science & Information Technology (CS & IT)
In equation (3), the first term intends to decrease reconstruction error, and the second term is used
to limit the overall ‘volume’ of simplex contrasted by endmembers. During the optimization
process, we can control the tradeoff between spectral reconstruction accuracy and the
distance/volume constraint ( )AJ via λ .
Furthermore, since abundance must satisfy sum-to-one constraints, we can adjust equation (5) by
adding 1M and 1N to endmember matrix A and original hyperspectral matrix X respectively.
To control the influence of sum-to-one constrains, we introduce a regulation factor α to 1M and
1N , as shown in Equation (6):
1 1
A X
A X
T T
M Nα α
← ←
(6)
4. SIMPLEX PATTERN AND PARAMETER SELECTION
Since algorithm’s performance fluctuated significantly with the variation of simplex pattern,
analysis of simplex pattern should be significant. Two constraints mentioned above differ in most
situations. It’s necessary to give a comparative analysis to decide which constraint is more
operative given certain simplex pattern.
4.1. Analysis of Simplex Pattern
As both VC and SSD can restrict simplex of endmember closing to original hyperspectral
scattering, it is necessary to analyze the equivalence for these two constraints. We mainly do
analysis in following two situations: First, for regular or quasi-regular simplex, there is a one-to-
one correspondence between volume and distance for simplex, as show in Figure 2(a). In this
situation, the VC and SSD have a similar performance.
However, when the simplex is not regular, namely non-regular simplex, the relationship between
volume and distance of simplex is indefinite, as shown in Figure 2(b). In this case, it is necessary
to analysis which one is better. Since in real hyperspectral data set, simplex for endmembers are
not always regular, analysis for non-regular case is very important. For convenience, we define η
as shown in (7) to measure degree of flatness in simplex.
=
θ
η
π
(7)
where, θ represents the maximum generalized angle of simplex.
Because of the inequivalence in non-regular simplex, flatness analysis is given to identify the
performance for each constraint. Additionally, to further differentiate VC and SSD, random
initialization and anti-noise analysis are implemented to compare algorithm performance.
5. Computer Science & Information Technology (CS & IT) 17
(a) regular simplex (b) non-regular simplex
Figure 2. Relationship between volume and distance for regular/non-regular simplex. In the case of regular
simplex, the relationship between two constraints is definite, while in the case of non-regular, the
relationship is indefinite.
4.2. Parameter Selection
To present a fair comparison, we need to guarantee all variables except for the constraint item to
be the same during unmixing process. For further details, (a) the update rule involved in these two
constraints is fixed to quadratic programming method and steepest descent method respectively.
(b) The regulation factor λ is carefully chosen to ensure similar ratio between constraint value
and reconstruction error in each case.
To decide Dλ and Vλ for each constraint, we need to find out the relationship between volume
V and sum of squared distance SSD for regular simplex. The relationship satisfies following
equation:
3
( ) MV
c M d
SSD
−
= (8)
where, M is the number of endmember, c is a variable only related to M , d is the distance
between any two endmembers.
Thus, in order to ensure equivalence of these two constraints, we need to make:
3
1 1
/ ( )
D
M
V V SSD c M d
λ
λ −
= = (9)
5. EXPERIMENT
In this part, we analyse application range of these two algorithms. Then we apply these two
algorithms to real hyperspectral data unmixing and compare the performance. As simplex pattern,
initial value and SNR are the most important factors in hyperspectral unmixing, we mainly
conduct the comparative analysis in these three aspects.
5.1. Comparison Criterion
In the process of comparing VC and SSD, a suitable comparison metric is needed to measure the
unmixing performance. Since, endmember data and abundance map can be transferred to
corresponding vector, we adopt angle distance(AD) which measures angular difference between
two vectors as criterion, as shown in equation (10). For spectral endmember and abundance map,
we refer to angle distance as spectral angle distance(SAD) and abundance angle distance(AAD)
separately.
6. 18 Computer Science & Information Technology (CS & IT)
cos ( )
|| |||| ||
x x
x x
T
SAD −
= 1 1 2
1 2
(10)
where, 1x and 2x are two transformed vectors. AAD is calculated in the similar way.
5.2. Analysis on Synthetic Data
For synthetic data, we pick several spectra from spectral library as endmembers. Then we create
hyperspectral data by multiplying normalized endmember data with abundance map generated
according to dirichlet distribution. Additionally, Gaussian noise with certain level is also added to
data.
1. Flatness Analysis
In this experiment, we reconstruct hyperspectral data with two bands and three endmembers. To
demonstrate unmixing capability, we increase the degree of flatness by certain value at each
experiment.
Then, we implement unmixing algorithm based on VC and SSD on hyperspectral data. The initial
values for both algorithms are randomly set. We compare experimental result of endmember
extraction with chosen spectra from library and compute SAD. The result is shown in figure3.
According to Figure 3, we can see that SAD of VC based algorithm is becoming increasingly
bigger compared to SSD with increase of degree of flatness. Whereas the resulting endmember
based on VC and SSD are similar when the simplex approaches to the regular form. Therefore,
we can draw the conclusion that SSD is better than VC when scattering is flat. In real data
processing, scattering on hyperplane is usually non-regular, so algorithm based on SSD can
handle most of these cases according to the result above.
0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95
0
1
2
3
4
5
6
7
Flatness Degree
SAD
SSD Constraint Volume Constraint
Figure 3 SAD of VC and SSD based algorithm.
2. Random Initialization Analysis
Normally, it is necessary to give an initial value. In this study, we used PPI or N-FINDR to obtain
a relatively suitable initial value for following iterations. For abundance matrix, it is often
initialized as random value. However, due to the complexity of real data set, these methods do not
always perform well on finding well-conditioned initial values. Consequently, the result may be
trapped into some local minima. Thus, random initial value analysis can give us a view that which
one is more susceptible to ill initial conditions. We carry out the same random initialization on
both VC and SSD algorithms, then we compare the result with original true endmember value.
7. Computer Science & Information Technology (CS & IT) 19
We use random values following Gaussian normal distribution as initial value of endmember and
abundance matrices. We conduct 50 comparison experiments with different initial values. Then
we compute SAD of resulting endmember data and endmember data in spectral library, as shown
in Figure 4.
We can see that endmember extraction result based on SSD is obviously closer to original
endmember data. However, some of the result based on VC absolutely deviate from original
endmember data. In several cases, though VC-based algorithm also can fit original hyperspectral
data well, the evaluated endmember data completely differs from original endmember and
consequently abundance matrix obtained by volume constraint is completely wrong.
0 5 10 15 20 25 30 35 40 45 50
0
2
4
6
8
10
12
14
16
Experimets
SAD
SSD Constraint Volume Constraint
Figure 4 Solution between two constrained algorithms with random initial values. Simulated hyperspectral
data consists of 5 bands and 6 endmembers.
3. Robustness Analysis for Noise
Since real data consists of much noise, original unmixing algorithm is sensitive to noise for the
sake of fitting every individual data sample. However, unmixing algorithms based on VC and
SSD can be applied to unmixing hyperspectral image by minimizing ‘volume’ of simplex. Thus,
reconstructed data will not inevitably approximate all data samples. As a result, these algorithms
show strong anti-noise capacity. However, as these two constraints are considered to be
inequivalent in many cases, noise-sensitivity may be different with each other. Furthermore, we
need to evaluate the suitable degree of SNR for VC and SSD.
We create 50 50× hyperspectral data including 5 bands and 6 endmembers. Then, we add white
noise with different levels to synthetic data. The SNR is 10db, 15db, 20db, 30db. During the
process of unmixing based on two constraints, we use identical iteration method with same upper
limit construction error. We implement two constrained algorithms under each SNR ratio with the
same initial value following Gaussian distribution during each experiment.
0 5 10 15 20 25 30 35 40 45 50
2
4
6
8
10
12
14
16
Experimets
SAD
SSD Constraint
Volume Constraint
(a) SAD with 10db SNR
0 5 10 15 20 25 30 35 40 45 50
2
4
6
8
10
12
14
16
Experimets
SAD
SSD Constraint
Volume Constraint
(b) SAD with 15db SNR
8. 20 Computer Science & Information Technology (CS & IT)
0 5 10 15 20 25 30 35 40 45 50
0
2
4
6
8
10
12
14
16
Experimets
SAD
SSD Constraint
Volume Constraint
(c) SAD with 20db SNR
0 5 10 15 20 25 30 35 40 45 50
0
2
4
6
8
10
12
14
Experimets
SAD
SSD Constraint
Volume Constraint
(d) SAD with 30db SNR
Figure 5 SAD for two constraints with different SNR
We can see from
Figure 5, SAD for VC between extracted and real endmember value change little with the decline
of SNR. On the contrary, SAD for SSD fluctuates with the SNR significantly. Thus, algorithm
based on VC is more robust than SSD in the sense of noise robustness.
5.3. Analysis on Real Data
After finishing the analysis above, we can conclude that SSD is better than VC when hyper
scattering with high degree of flatness or under ill-conditioned initial values. While VC is better
in the sense of robustness to noise. However, above-mentioned experiments are based on
synthetic data. In this experiment, we will utilize real data(AVIRIS data) to identify these two
algorithms.
The used AVIRIS data over Cuprite, Nevada totally contains 400*350 pixels and 50 bands. We
do some pre-processes to raw data to reduce computation complexity before iteration. Firstly, we
utilize principal component analysis (PCA) [16] to reduce data dimension and select principal
band numbers. Secondly, we need to find good initial endmember and abundance value to ensure
algorithms can extract real ground objects efficiently. We utilize endmember data extracted by N-
FINDR as initial endmember matrix. Since hyperspectral data can be regarded as the product of
endmember matrix and abundance matrix, we use unconstrained NMF algorithm to calculate
corresponding abundance matrix S as initial value by fixing endmember abundance A .In
addition, experiment shows that we can achieve much better results by assigning regulation factor
Vλ as 0.15 for VC and Dλ as 0.01 for SSD.
Table 1 SAD among different algorithm
N-FINDR Volume Constraint SSD Constraint
Alunite 4.43 5.45 4.00
Kaolinite 3.28 5.29 5.35
Andradite 4.41 5.03 4.35
Nontronite 4.14 7.58 4.19
Muscovite 6.16 2.34 4.71
Chalcedony 3.75 6.86 3.57
Average 4.36 5.43 4.46
Then, we begin to do iteration for two constraint algorithms until it satisfies terminating
condition. we can find out best matching mineral obtained by two algorithms via comparing with
each mineral reflectance in spectral library [17].
9. Computer Science & Information Technology (CS & IT) 21
As shown in Figure 6, it’s the endmember extraction result based on SSD. Solid line and dashed
line represent extraction result obtained by SSD and best matched data in spectral library
respectively. The reflectance is normalized to [0, 1]. From several subgraph results, endmembers
extracted by SSD are very close to the spectral of real data, like Alunite and Muscovite. The
closer they are, the smaller SAD is. Table 1 represents SAD for different algorithms. For some
minerals, SSD gives a better solution. For other minerals, VC performs better.
Figure 6 Extracted endmember by SSD.
6. CONCLUSION
In this paper, we analyse VC and SSD algorithm from flatness of simplex, anti-noise and
initialization to discriminate these two algorithms. We aim to provide a guidance on which
constraint is more suitable under some special conditions.
First, we do analysis for flatness and conduct three comparative experiments using synthetic data.
For the pattern of scattering, SSD is better than VC when the scattering is flat. Whereas these two
algorithms’ performance resemble each other while the simplex is regular. As for initialization,
endmember extracted under SSD is closer to original data in random initialization. For anti-noise
performance, VC is more robust in different level of noise.
Eventually, on real data, similar solutions can be achieved for these two constrains with well-
conditioned initial value. Quantitively, for some minerals, SAD of SSD is smaller, like
Chalcedony in Table 1. Yet, for other minerals, like Muscovite, VC works better. Thus, VC and
SSD both work similarly in hyperspectral unmixing task.
According to what mentioned above, relatively practical instruction on how to choose constraints
can be attained.
10. 22 Computer Science & Information Technology (CS & IT)
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