This book provides an overview of existing models, methods and algorithms as an introduction to the topic, and presents several new techniques and approaches developed by the authors that can be applied to non-negative matrix and tensor factorizations. The authors focus on algorithms which are fast and robust in the field of NMFs, as well as related models which have much flexibility, and therefore are the most useful in practice. Using generalized cost functions such as Bregman, alpha and beta divergences, the authors present practical implementations of several types of robust algorithms, in particular Multiplicative, Projected Gradient and Quasi Newton algorithms. A comparative analysis of the different methods is given in order to identify approximation error and complexity Single source reference guide to NMF, collating information that is widely dispersed in current literature, such as in conference papers, journal articles and tutorials Provides an overview of existing models, methods and algorithms, as well as presenting new techniques and approaches developed by the authors Written by leading researchers in the relatively new but maturing area of non-negative matrix factorizations (NMFs) within blind signal processing Includes support material on a website which contains MATLAB toolboxes

Andrzej Cichocki, Laboratory for Advanced Brain Signal Processing, Riken Brain Science Institute, Japan
Professor Cichocki is head of the Laboratory for Advanced Brain Signal Processing. He has co-authored more than one hundred technical papers, and is the author of three previous books of which two are published by Wiley. His most recent book is Adaptive Blind Signal and Image Processing with Professor Shun-ichi Amari (Wiley, 2002). He is Editor-in-Chief of International Journal Computational Intelligence and Neuroscience and Associate Editor of IEEE Transactions on Neural Networks.

Shun-ichi Amari, Laboratory for Mathematical Neuroscience, Riken Brain Science Institute, Japan
Professor Amari is head of the Laboratory for Mathematical Neuroscience, as well as vice-president of the Riken Brain Science Institute. He serves on editorial boards for numerous journals including Applied Intelligence, Journal of Mathematical Systems and Control and Annals of Institute of Statistical Mathematics. He is the co-author of three books, and more than three hundred technical papers.

Rafal Zdunek, Institute of Telecommunications, Teleinformatics and Acoustics, Wroclaw University of Technology, Poland

Associate Professor Zdunek is currently a lecturer at the Wroclaw University of Technology, Poland and up until recently was a visiting research scientist at the Riken Brain Science Institute. He is a member of the IEEE: Signal Processing Society, Communications Society and a member of the Society of Polish Electrical Engineers. Dr Zdunek has guest co-edited with Professor Cichocki amongst others, a special issue on Advances in Non-negative Matrix and Tensor Factorization in the journal, Computational Intelligence and Neuroscience (published May 08).

Anh Huy Phan, Laboratory for Advanced Brain Signal Processing, Riken Brain Science Institute, Japan
Anh Huy Phan is a researcher at the Laboratory for Advanced Brian Signal Processing at the Riken Brain Science Institute.

Preface
Acknowledgments
Glossary of Symbols and Abbreviations
Introduction - Problem Statements and Models
Blind Source Separation and Linear Generalized Component Analysis
Matrix Factorization Models with Nonnegativity and Sparsity Constraints
Why Nonnegativity and Sparsity Constraints?
Basic NMF Model
Symmetric NMF
Semi-Orthogonal NMF
Semi-NMF and Nonnegative Factorization of Arbitrary Matrix
Three-factor NMF
NMF with Offset
Multi-layer NMF
Simultaneous NMF
Projective and Convex NMF
Kernel NMF
Convolutive NMF
Overlapping NMF
Basic Approaches to Estimate Parameters of Standard NMF
Large-scale NMF
Non-uniqueness of NMF and Techniques to Alleviate the Ambiguity Problem
Initialization of NMF
Stopping Criteria
Tensor Properties and Basis of Tensor Algebra
Tensors (Multi-way Arrays) - Preliminaries
Subarrays, Tubes and Slices
Unfolding - Matricization
Vectorization
Outer, Kronecker, Khatri-Rao and Hadamard Products
Mode-n Multiplication of Tensor by Matrix and Tensor by Vector, Contracted Tensor Product
Special Forms of Tensors
Tensor Decompositions and Factorizations
Why Multi-way Array Decompositions and Factorizations?
PARAFAC and Nonnegative Tensor Factorization
NTF1 Model
NTF2 Model
Individual Differences in Scaling (INDSCAL) and Implicit Slice Canonical Decomposition Model (IMCAND)
Shifted PARAFAC and Convolutive NTF
Nonnegative Tucker Decompositions
Block Component Decompositions
Block-Oriented Decompositions
Paratuck2 And Dedicom Models
Hierarchical Tensor Decomposition
Discussion and Conclusions
Similarity Measures and Generalized Divergences
Error-induced Distance and Robust Regression Techniques
Robust Estimation
Csiszár Divergences
Bregman Divergence
Bregman Matrix Divergences
Alpha-Divergences
Asymmetric Alpha-Divergences
Symmetric Alpha-Divergences
Beta-Divergences
Gamma-Divergences
Divergences Derived from Tsallis and Rényi Entropy
Concluding Remarks
Multiplicative Iterative Algorithms for NMF with Sparsity Constraints
Extended ISRA and EMML Algorithms: Regularization and Sparsity
Multiplicative NMF Algorithms Based on the Squared Euclidean Distance
Multiplicative NMF Algorithms Based on Kullback-Leibler I-Divergence
Multiplicative Algorithms Based on Alpha-Divergence
Multiplicative Alpha NMF Algorithm
Generalized Multiplicative Alpha NMF Algorithms
Alternating SMART: Simultaneous Multiplicative Algebraic Reconstruction Technique
Alpha SMART Algorithm
Generalized SMART Algorithms
Multiplicative NMF Algorithms Based on Beta-Divergence
Multiplicative Beta NMF Algorithm
Multiplicative Algorithm Based on the Itakura-Saito Distance
Generalized Multiplicative Beta Algorithm for NMF
Algorithms for Semi-orthogonal NMF and Orthogonal Three-Factor NMF
Multiplicative Algorithms for Affine NMF
Multiplicative Algorithms for Convolutive NMF
Multiplicative Algorithm for Convolutive NMF Based on Alpha-Divergence
Multiplicative Algorithm for Convolutive NMF Based on Beta-Divergence
Efficient Implementation of CNMF Algorithm
Simulation Examples for Standard NMF
Examples for Affine NMF
Music Analysis and Decomposition Using Convolutive NMF
Discussion and Conclusions
Alternating Least Squares and Related Algorithms for NMF and SCA Problems
Standard ALS Algorithm
Multiple Linear Regression - Vectorized Version of ALS Update Formulas
Weighted ALS
Methods for Improving Performance and Convergence Speed of ALS Algorithms
ALS Algorithm for Very Large-scale NMF
ALS Algorithm with Line-Search
Acceleration of ALS Algorithm via Simple Regularization
ALS Algorithm with Flexible and Generalized Regularization Terms
ALS with Tikhonov Type Regularization Terms
ALS Algorithms with Sparsity Control and Decorrelation
Combined Generalized Regularized ALS Algorithms
Wang-Hancewicz Modified ALS Algorithm
Implementation of Regularized ALS Algorithms for NMF
HALS Algorithm and its Extensions
Projected Gradient Local Hierarchical Alternating Least Squares (HALS) Algorithm
Extensions and Implementations of the HALS Algorithm
Fast HALS NMF Algorithm for Large-scale Problems
HALS NMF Algorithm with Sparsity, Smoothness and Uncorrelatedness Constraints
HALS Algorithm for Sparse Component Analysis and Flexible Component Analysis
Simplified HALS Algorithm for Distributed and Multi-task Compressed Sensing
Generalized HALS-CS Algorithm
Generalized HALS Algorithms Using Alpha-Divergence
Generalized HALS Algorithms Using Beta-Divergence
Simulation Results
Underdetermined Blind Source Separation Examples
NMF with Sparseness, Orthogonality and Smoothness Constraints
Simulations for Large-scale NMF
Illustrative Examples for Compressed Sensing
Discussion and Conclusions
Projected Gradient Algorithms
Oblique Projected Landweber (OPL) Method
Lin's Projected Gradient (LPG) Algorithm with Armijo Rule
Barzilai-Borwein Gradient Projection for Sparse Reconstruction (GPSR-BB)
Projected Sequential Subspace Optimization (PSESOP)
Interior Point Gradient (IPG) Algorithm
Interior Point Newton (IPN) Algorithm
Regularized Minimal Residual Norm Steepest Descent Algorithm (RMRNSD)
Sequential Coordinate-Wise Algorithm (SCWA)
Simulations
Discussions
Quasi-Newton Algorithms for Nonnegative Matrix Factorization
Projected Quasi-Newton Optimization
Projected Quasi-Newton for Frobenius Norm
Projected Quasi-Newton for Alpha-Divergence
Projected Quasi-Newton for Beta-Divergence
Practical Implementation
Gradient Projection Conjugate Gradient
FNMA algorithm
NMF with Quadratic Programming
Nonlinear Programming
Quadratic Programming
Trust-region Subproblem
Updates for A
Hybrid Updates
Numerical Results
Discussions
Multi-Way Array (Tensor) Factorizations and Decompositions
Learning Rules for the Extended Three-way NTF1 Problem
Basic Approaches for the Extended NTF1 Model
ALS Algorithms for NTF1
Multiplicative Alpha and Beta Algorithms for the NTF1 Model
Multi-layer NTF1 Strategy
Algorithms for Three-way Standard and Super Symmetric Nonnegative Tensor Factorization
Multiplicative NTF Algorithms Based on Alpha- and Beta-Divergences
Simple Alternative Approaches for NTF and SSNTF
Nonnegative Tensor Factorizations for Higher-Order Arrays
Alpha NTF Algorithm
Beta NTF Algorithm
Fast HALS NTF Algorithm Using Squared Euclidean Distance
Generalized HALS NTF Algorithms Using Alpha- and Beta-Divergences
Tensor Factorization with Additional Constraints
Algorithms for Nonnegative and Semi-Nonnegative Tucker Decompositions
Higher Order SVD (HOSVD) and Higher Order Orthogonal Iteration (HOOI) Algorithms
ALS Algorithm for Nonnegative Tucker Decomposition
HOSVD, HOOI and ALS Algorithms as Initialization Tools for Nonnegative Tensor Decomposition
Multiplicative Alpha Algorithms for Nonnegative Tucker Decomposition
Beta NTD Algorithm
Local ALS Algorithms for Nonnegative TUCKER Decompositions
Semi-Nonnegative Tucker Decomposition
Nonnegative Block-Oriented Decomposition
Multiplicative Algorithms for NBOD
Multi-level Nonnegative Tensor Decomposition - High Accuracy Compression and Approximation
Simulations and Illustrative Examples
Experiments for Nonnegative Tensor Factorizations
Experiments for Nonnegative Tucker Decomposition
Experiments for Nonnegative Block-Oriented Decomposition
Multi-Way Analysis of High Density Array EEG - Classification of Event Related Potentials
Application of Tensor Decompositions in Brain Computer Interface - Classification of Motor Imagery Tasks
Image and Video Applications
Discussion and Conclusions
Selected Applications
Clustering
Semi-Binary NMF
NMF vs. Spectral Clustering
Clustering with Convex NMF
Application of NMF to Text Mining
Email Surveillance
Classification
Musical Instrument Classification
Image Classification
Spectroscopy
Raman Spectroscopy
Fluorescence Spectroscopy
Hyperspectral Imaging
Chemical Shift Imaging
Application of NMF for Analyzing Microarray Data
Gene Expression Classification
Analysis of Time Course Microarray Data
References
Index
Table of Contents provided by Publisher. All Rights Reserved.

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