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Nonnegative matrix and tensor factorizations : applications to exploratory multi-way data analysis and blind source separation
- ISBN:9780470746660
- ISBN:0470746661
- Call Number : QA 76 .9 .A43 .N65 2009
- Title:Nonnegative matrix and tensor factorizations : applications to exploratory multi-way data analysis and blind source separation [electronic resource] / Andrzej Cichocki ... [et al.].
- Publisher:Chichester, U.K. : John Wiley, 2009.
- Physical Description:xxi, 477 p. : col. ill. ; 25 cm
- Notes:Includes bibliographical references and index
- Subject:Computer algorithms.
- Subject:Data mining.
- Subject:Machine learning.
- Subject:Data structures (Computer science)
- Added Entry:Cichocki, Andrzej.
Nonnegative matrix and tensor factorizationsمحتواي کتاب
- NONNEGATIVE MATRIX AND TENSOR FACTORIZATIONS: APPLICATIONS TO EXPLORATORY MULTI-WAY DATA ANALYSIS AND BLIND SOURCE SEPARATION
- Contents
- Preface
- Acknowledgments
- Glossary of Symbols and Abbreviations
- 1 Introduction - Problem Statements and Models
- 1.1 Blind Source Separation and Linear Generalized Component Analysis
- 1.2 Matrix Factorization Models with Nonnegativity and Sparsity Constraints
- 1.2.1 Why Nonnegativity and Sparsity Constraints?
- 1.2.2 Basic NMF Model
- 1.2.3 Symmetric NMF
- 1.2.4 Semi-Orthogonal NMF
- 1.2.5 Semi-NMF and Nonnegative Factorization of Arbitrary Matrix
- 1.2.6 Three-factor NMF
- 1.2.7 NMF with Offset (Affine NMF)
- 1.2.8 Multi-layer NMF
- 1.2.9 Simultaneous NMF
- 1.2.10 Projective and Convex NMF
- 1.2.11 Kernel NMF
- 1.2.12 Convolutive NMF
- 1.2.13 Overlapping NMF
- 1.3 Basic Approaches to Estimate Parameters of Standard NMF
- 1.4 Tensor Properties and Basis of Tensor Algebra
- 1.4.1 Tensors (Multi-way Arrays) – Preliminaries
- 1.4.2 Subarrays, Tubes and Slices
- 1.4.3 Unfolding – Matricization
- 1.4.4 Vectorization
- 1.4.5 Outer, Kronecker, Khatri-Rao and Hadamard Products
- 1.4.6 Mode-n Multiplication of Tensor by Matrix and Tensor by Vector, Contracted Tensor Product
- 1.4.7 Special Forms of Tensors
- 1.5 Tensor Decompositions and Factorizations
- 1.5.1 Why Multi-way Array Decompositions and Factorizations?
- 1.5.2 PARAFAC and Nonnegative Tensor Factorization
- 1.5.3 NTF1 Model
- 1.5.4 NTF2 Model
- 1.5.5 Individual Differences in Scaling (INDSCAL) and Implicit Slice Canonical Decomposition Model (IMCAND)
- 1.5.6 Shifted PARAFAC and Convolutive NTF
- 1.5.7 Nonnegative Tucker Decompositions
- 1.5.8 Block Component Decompositions
- 1.5.9 Block-Oriented Decompositions
- 1.5.10 PARATUCK2 and DEDICOM Models
- 1.5.11 Hierarchical Tensor Decomposition
- 1.6 Discussion and Conclusions
- Appendix 1.A: Uniqueness Conditions for Three-way Tensor Factorizations
- Appendix 1.B: Singular Value Decomposition (SVD) and Principal Component Analysis (PCA) with Sparsity and/or Nonnegativity Constraints
- Appendix 1.C: Determining a True Number of Components
- Appendix 1.D: Nonnegative Rank Factorization Using Wedderborn Theorem – Estimation of the Number of Components
- References
- 2 Similarity Measures and Generalized Divergences
- 2.1 Error-induced Distance and Robust Regression Techniques
- 2.2 Robust Estimation
- 2.3 Csiszár Divergences
- 2.4 Bregman Divergence
- 2.5 Alpha-Divergences
- 2.6 Beta-Divergences
- 2.7 Gamma-Divergences
- 2.8 Divergences Derived from Tsallis and Rényi Entropy
- Appendix 2.A: Information Geometry, Canonical Divergence, and Projection
- Appendix 2.B: Probability Density Functions for Various Distributions
- References
- 3 Multiplicative Iterative Algorithms for NMF with Sparsity Constraints
- 3.1 Extended ISRA and EMML Algorithms: Regularization and Sparsity
- 3.2 Multiplicative Algorithms Based on Alpha-Divergence
- 3.3 Alternating SMART: Simultaneous Multiplicative Algebraic Reconstruction Technique
- 3.4 Multiplicative NMF Algorithms Based on Beta-Divergence
- 3.5 Algorithms for Semi-orthogonal NMF and Orthogonal Three-Factor NMF
- 3.6 Multiplicative Algorithms for Affine NMF
- 3.7 Multiplicative Algorithms for Convolutive NMF
- 3.8 Simulation Examples for Standard NMF
- 3.9 Examples for Affine NMF
- 3.10 Music Analysis and Decomposition Using Convolutive NMF
- 3.11 Discussion and Conclusions
- Appendix 3.A: Fast Algorithms for Large-scale Data
- Appendix 3.B: Performance Evaluation
- Appendix 3.C: Convergence Analysis of the Multiplicative Alpha NMF Algorithm
- Appendix 3.D: MATLAB Implementation of the Multiplicative NMF Algorithms
- Appendix 3.E: Additional MATLAB Functions
- References
- 4 Alternating Least Squares and Related Algorithms for NMF and SCA Problems
- 4.1 Standard ALS Algorithm
- 4.2 Methods for Improving Performance and Convergence Speed of ALS Algorithms
- 4.3 ALS Algorithm with Flexible and Generalized Regularization Terms
- 4.4 Combined Generalized Regularized ALS Algorithms
- 4.5 Wang-Hancewicz Modified ALS Algorithm
- 4.6 Implementation of Regularized ALS Algorithms for NMF
- 4.7 HALS Algorithm and its Extensions
- 4.7.1 Projected Gradient Local Hierarchical Alternating Least Squares (HALS) Algorithm
- 4.7.2 Extensions and Implementations of the HALS Algorithm
- 4.7.3 Fast HALS NMF Algorithm for Large-scale Problems
- 4.7.4 HALS NMF Algorithm with Sparsity, Smoothness and Uncorrelatedness Constraints
- 4.7.5 HALS Algorithm for Sparse Component Analysis and Flexible Component Analysis
- 4.7.6 Simplified HALS Algorithm for Distributed and Multi-task Compressed Sensing
- 4.7.7 Generalized HALS-CS Algorithm
- 4.7.8 Generalized HALS Algorithms Using Alpha-Divergence
- 4.7.9 Generalized HALS Algorithms Using Beta-Divergence
- 4.8 Simulation Results
- 4.9 Discussion and Conclusions
- Appendix 4.A: MATLAB Source Code for ALS Algorithm
- Appendix 4.B: MATLAB Source Code for Regularized ALS Algorithms
- Appendix 4.C: MATLAB Source Code for Mixed ALS-HALS Algorithms
- Appendix 4.D: MATLAB Source Code for HALS CS Algorithm
- Appendix 4.E: Additional MATLAB Functions
- References
- 5 Projected Gradient Algorithms
- 5.1 Oblique Projected Landweber (OPL) Method
- 5.2 Lin’s Projected Gradient (LPG) Algorithm with Armijo Rule
- 5.3 Barzilai-Borwein Gradient Projection for Sparse Reconstruction (GPSR-BB)
- 5.4 Projected Sequential Subspace Optimization (PSESOP)
- 5.5 Interior Point Gradient (IPG) Algorithm
- 5.6 Interior Point Newton (IPN) Algorithm
- 5.7 Regularized Minimal Residual Norm Steepest Descent Algorithm (RMRNSD)
- 5.8 Sequential Coordinate-Wise Algorithm (SCWA)
- 5.9 Simulations
- 5.10 Discussions
- Appendix 5.A: Stopping Criteria
- Appendix 5.B: MATLAB Source Code for Lin’s PG Algorithm
- References
- 6 Quasi-Newton Algorithms for Nonnegative Matrix Factorization
- 7 Multi-Way Array (Tensor) Factorizations and Decompositions
- 7.1 Learning Rules for the Extended Three-way NTF1 Problem
- 7.2 Algorithms for Three-way Standard and Super Symmetric Nonnegative Tensor Factorization
- 7.3 Nonnegative Tensor Factorizations for Higher-Order Arrays
- 7.4 Algorithms for Nonnegative and Semi-Nonnegative Tucker Decompositions
- 7.4.1 Higher Order SVD (HOSVD) and Higher Order Orthogonal Iteration (HOOI) Algorithms
- 7.4.2 ALS Algorithm for Nonnegative Tucker Decomposition
- 7.4.3 HOSVD, HOOI and ALS Algorithms as Initialization Tools for Nonnegative Tensor Decomposition
- 7.4.4 Multiplicative Alpha Algorithms for Nonnegative Tucker Decomposition
- 7.4.5 Beta NTD Algorithm
- 7.4.6 Local ALS Algorithms for Nonnegative TUCKER Decompositions
- 7.4.7 Semi-Nonnegative Tucker Decomposition
- 7.5 Nonnegative Block-Oriented Decomposition
- 7.6 Multi-level Nonnegative Tensor Decomposition - High Accuracy Compression and Approximation
- 7.7 Simulations and Illustrative Examples
- 7.7.1 Experiments for Nonnegative Tensor Factorizations
- 7.7.2 Experiments for Nonnegative Tucker Decomposition
- 7.7.3 Experiments for Nonnegative Block-Oriented Decomposition
- 7.7.4 Multi-Way Analysis of High Density Array EEG – Classification of Event Related Potentials
- 7.7.5 Application of Tensor Decomposition in Brain Computer Interface – Classification of Motor Imagery Tasks
- 7.7.6 Image and Video Applications
- 7.8 Discussion and Conclusions
- Appendix 7.A: Evaluation of Interactions and Relationships Among Hidden Components for NTD Model
- Appendix 7.B: Computation of a Reference Tensor
- Appendix 7.C: Trilinear and Direct Trilinear Decompositions for Efficient Initialization
- Appendix 7.D: MATLAB Source Code for Alpha NTD Algorithm
- Appendix 7.E: MATLAB Source Code for Beta NTD Algorithm
- Appendix 7.F: MATLAB Source Code for HALS NTD Algorithm
- Appendix 7.G: MATLAB Source Code for ALS NTF1 Algorithm
- Appendix 7.H: MATLAB Source Code for ISRA BOD Algorithm
- Appendix 7.I: Additional MATLAB functions
- References
- 8 Selected Applications
- Index