Nonnegative Matrix and Tensor Factorization T

This book provides a broad survey of models and efficient algorithms for Nonnegative Matrix Factorization (NMF). This includes NMFs various extensions and modifications, especially Nonnegative Tensor Factorizations (NTF) and Nonnegative Tucker Decompositions (NTD). NMF/NTF and their extensions are increasingly used as tools in signal and image processing, and data analysis, having garnered interest due to their capability to provide new insights and relevant information about the complex latent relationships in experimental data sets. It is suggested that NMF can provide meaningful components with physical interpretations; for example, in bioinformatics, NMF and its extensions have been successfully applied to gene expression, sequence analysis, the functional characterization of genes, clustering and text mining. As such, the authors focus on the algorithms that are most useful in practice, looking at the fastest, most robust, and suitable for large-scale models. Key features: Acts as a single source reference guide to NMF, collating information that is widely dispersed in current literature, including the authors own recently developed techniques in the subject area. Uses generalized cost functions such as Bregman, Alpha and Beta divergences, to present practical implementations of several types of robust algorithms, in particular Multiplicative, Alternating Least Squares, Projected Gradient and Quasi Newton algorithms. Provides a comparative analysis of the different methods in order to identify approximation error and complexity. Includes pseudo codes and optimized MATLAB source codes for almost all algorithms presented in the book. The increasing interest in nonnegative matrix and tensor factorizations, as well as decompositions and sparse representation of data, will ensure that this book is essential reading for engineers, scientists, researchers, industry practitioners and graduate students across signal and image processing; neuroscience; data mining and data analysis; computer science; bioinformatics; speech processing; biomedical engineering; and multimedia.

[1]  Paul Van Dooren,et al.  Descent methods for Nonnegative Matrix Factorization , 2008, ArXiv.

[2]  R. Harshman,et al.  Shifted factor analysis—Part I: Models and properties , 2003 .

[3]  S. Goreinov,et al.  A Theory of Pseudoskeleton Approximations , 1997 .

[4]  Yaakov Tsaig,et al.  Extensions of compressed sensing , 2006, Signal Process..

[5]  R. Bro,et al.  PARAFAC and missing values , 2005 .

[6]  Pierre Comon,et al.  Tensor Decompositions, State of the Art and Applications , 2002 .

[7]  Mohamed-Jalal Fadili,et al.  Morphological Diversity and Sparsity in Blind Source Separation , 2007, ICA.

[8]  Andrzej Cichocki,et al.  New Algorithms for Non-Negative Matrix Factorization in Applications to Blind Source Separation , 2006, 2006 IEEE International Conference on Acoustics Speech and Signal Processing Proceedings.

[9]  Yuanqing Li,et al.  Blind estimation of channel parameters and source components for EEG signals: a sparse factorization approach , 2006, IEEE Transactions on Neural Networks.

[10]  Jianhua Z. Huang,et al.  Sparse principal component analysis via regularized low rank matrix approximation , 2008 .

[12]  A. Stegeman.  On uniqueness conditions for Candecomp/Parafac and Indscal with full column rank in one mode , 2009 .

[13]  Manfred K. Warmuth,et al.  Exponentiated Gradient Versus Gradient Descent for Linear Predictors , 1997, Inf. Comput..

[14]  Andrzej Cichocki,et al.  Hierarchical ALS Algorithms for Nonnegative Matrix and 3D Tensor Factorization , 2007, ICA.

[15]  D. Rubin,et al.  Maximum likelihood from incomplete data via the EM - algorithm plus discussions on the paper , 1977 .

[16]  H. Law.  Research methods for multimode data analysis , 1984 .

[17]  Jean-Luc Starck,et al.  Compressed Sensing in Astronomy , 2008, IEEE Journal of Selected Topics in Signal Processing.

[18]  Inderjit S. Dhillon,et al.  A generalized maximum entropy approach to bregman co-clustering and matrix approximation , 2004, J. Mach. Learn. Res..

[19]  Tamara G. Kolda,et al.  Tensor Decompositions and Applications , 2009, SIAM Rev..

[20]  Michael Elad,et al.  Optimized Projections for Compressed Sensing , 2007, IEEE Transactions on Signal Processing.

[21]  Fumikazu Miwakeichi,et al.  Decomposing EEG data into space–time–frequency components using Parallel Factor Analysis , 2004, NeuroImage.

[22]  Rasmus Bro,et al.  Multi-way Analysis with Applications in the Chemical Sciences , 2004 .

[23]  N. Čencov.  Statistical Decision Rules and Optimal Inference , 2000 .

[24]  Daniel M. Dunlavy,et al.  An Optimization Approach for Fitting Canonical Tensor Decompositions. , 2009 .

[25]  Liqing Zhang,et al.  A Note on Lewicki-Sejnowski Gradient for Learning Overcomplete Representations , 2008, Neural Computation.

[26]  Nicolas Gillis,et al.  Nonnegative Factorization and The Maximum Edge Biclique Problem , 2008, 0810.4225.

[27]  Hualou Liang,et al.  Single-Trial Decoding of Bistable Perception Based on Sparse Nonnegative Tensor Decomposition , 2008, Comput. Intell. Neurosci..

[28]  Michael W. Berry,et al.  Algorithms and applications for approximate nonnegative matrix factorization , 2007, Comput. Stat. Data Anal..

[29]  C. D. Meyer,et al.  Initializations for the Nonnegative Matrix Factorization , 2006 .

[30]  R. E. Cline,et al.  The generalized inverse of a nonnegative matrix , 1972 .

[31]  Pablo Tamayo,et al.  Metagenes and molecular pattern discovery using matrix factorization , 2004, Proceedings of the National Academy of Sciences of the United States of America.

[32]  Tamir Hazan,et al.  Sparse image coding using a 3D non-negative tensor factorization , 2005, Tenth IEEE International Conference on Computer Vision (ICCV'05) Volume 1.

[33]  Narendra Ahuja,et al.  Compact representation of multidimensional data using tensor rank-one decomposition , 2004, Proceedings of the 17th International Conference on Pattern Recognition, 2004. ICPR 2004..

[34]  Age K. Smilde,et al.  Constrained three‐mode factor analysis as a tool for parameter estimation with second‐order instrumental data , 1998 .

[35]  J. Kruskal,et al.  How 3-MFA data can cause degenerate parafac solutions, among other relationships , 1989 .

[36]  Hualiang Li,et al.  Non-negative Matrix Factorization with Orthogonality Constraints and its Application to Raman Spectroscopy , 2007, J. VLSI Signal Process..

[37]  Lieven De Lathauwer,et al.  A Link between the Canonical Decomposition in Multilinear Algebra and Simultaneous Matrix Diagonalization , 2006, SIAM J. Matrix Anal. Appl..

[38]  Alvaro R. De Pierro,et al.  A row-action alternative to the EM algorithm for maximizing likelihood in emission tomography , 1996, IEEE Trans. Medical Imaging.

[39]  P. Paatero,et al.  Positive matrix factorization: A non-negative factor model with optimal utilization of error estimates of data values† , 1994 .

[40]  David L. Donoho,et al.  WaveLab and Reproducible Research , 1995 .

[41]  Shun-ichi Amari,et al.  Dualistic geometry of the manifold of higher-order neurons , 1991, Neural Networks.

[42]  Narendra Ahuja,et al.  A Tensor Approximation Approach to Dimensionality Reduction , 2008, International Journal of Computer Vision.

[43]  L. Breiman.  Heuristics of instability and stabilization in model selection , 1996 .

[44]  H. Kiers,et al.  Three-mode principal components analysis: choosing the numbers of components and sensitivity to local optima. , 2000, The British journal of mathematical and statistical psychology.

[45]  Lieven De Lathauwer,et al.  Decompositions of a Higher-Order Tensor in Block Terms - Part II: Definitions and Uniqueness , 2008, SIAM J. Matrix Anal. Appl..

[46]  L. Lathauwer,et al.  Sufficient conditions for uniqueness in Candecomp/Parafac and Indscal with random component matrices , 2006, Psychometrika.

[47]  William S Rayens,et al.  Structure-seeking multilinear methods for the analysis of fMRI data , 2004, NeuroImage.

[48]  Christos Boutsidis,et al.  SVD based initialization: A head start for nonnegative matrix factorization , 2008, Pattern Recognit..

[49]  T. Hebert,et al.  A generalized EM algorithm for 3-D Bayesian reconstruction from Poisson data using Gibbs priors. , 1989, IEEE transactions on medical imaging.

[50]  Judith C. Brown.  Calculation of a constant Q spectral transform , 1991 .

[51]  Yizhou Yu,et al.  Hierarchical Tensor Approximation of Multidimensional Images , 2007, 2007 IEEE International Conference on Image Processing.

[52]  I. Daubechies,et al.  An iterative thresholding algorithm for linear inverse problems with a sparsity constraint , 2003, math/0307152.

[53]  H. Kiers,et al.  Discriminating between strong and weak structures in three-mode principal component analysis. , 2009, The British journal of mathematical and statistical psychology.

[54]  Michael Elad,et al.  Automatic parameter setting for iterative shrinkage methods , 2008, 2008 IEEE 25th Convention of Electrical and Electronics Engineers in Israel.

[55]  D. Donoho,et al.  Translation-Invariant De-Noising , 1995 .

[56]  S. M. Ali,et al.  A General Class of Coefficients of Divergence of One Distribution from Another , 1966 .

[57]  Rasmus Bro,et al.  The N-way Toolbox for MATLAB , 2000 .

[58]  Shun-ichi Amari,et al.  Adaptive blind signal processing-neural network approaches , 1998, Proc. IEEE.

[59]  Andrzej Cichocki,et al.  Csiszár's Divergences for Non-negative Matrix Factorization: Family of New Algorithms , 2006, ICA.

[60]  Victor Solo,et al.  Dimension Estimation in Noisy PCA With SURE and Random Matrix Theory , 2008, IEEE Transactions on Signal Processing.

[61]  Tamara G. Kolda,et al.  Temporal Analysis of Semantic Graphs Using ASALSAN , 2007, Seventh IEEE International Conference on Data Mining (ICDM 2007).

[62]  Andrzej Cichocki,et al.  Robust techniques for independent component analysis (ICA) with noisy data , 1998, Neurocomputing.

[63]  Tamir Hazan,et al.  Multi-way Clustering Using Super-Symmetric Non-negative Tensor Factorization , 2006, ECCV.

[64]  Lieven De Lathauwer,et al.  Decompositions of a Higher-Order Tensor in Block Terms - Part III: Alternating Least Squares Algorithms , 2008, SIAM J. Matrix Anal. Appl..

[65]  Richard A. Harshman,et al.  Foundations of the PARAFAC procedure: Models and conditions for an "explanatory" multi-model factor analysis , 1970 .

[66]  P. Anandan,et al.  Hierarchical Model-Based Motion Estimation , 1992, ECCV.

[67]  Vincent N. LaRiccia,et al.  Maximum Smoothed Likelihood Density Estimation for Inverse Problems , 1995 .

[68]  E. Oja,et al.  Independent Component Analysis , 2013 .

[69]  Michel E. B. Yamagishi,et al.  Fast iterative methods applied to tomography models with general Gibbs priors , 1999, Optics & Photonics.

[70]  Ralf Sarlette,et al.  Efficient and Realistic Visualization of Cloth , 2003, Rendering Techniques.

[71]  Rasmus Bro,et al.  Improving the speed of multiway algorithms: Part II: Compression , 1998 .

[72]  Lieven De Lathauwer,et al.  Decompositions of a Higher-Order Tensor in Block Terms - Part I: Lemmas for Partitioned Matrices , 2008, SIAM J. Matrix Anal. Appl..

[73]  B. Silverman,et al.  Wavelet thresholding via a Bayesian approach , 1998 .

[74]  Ali Ghodsi,et al.  Nonnegative matrix factorization via rank-one downdate , 2008, ICML '08.

[75]  B. Kowalski,et al.  Tensorial resolution: A direct trilinear decomposition , 1990 .

[76]  Petros Drineas,et al.  CUR matrix decompositions for improved data analysis , 2009, Proceedings of the National Academy of Sciences.

[77]  Lawrence Carin,et al.  Bayesian Compressive Sensing , 2008, IEEE Transactions on Signal Processing.

[78]  Zhaoshui He,et al.  Extended SMART Algorithms for Non-negative Matrix Factorization , 2006, ICAISC.

[79]  Minje Kim,et al.  Monaural Music Source Separation: Nonnegativity, Sparseness, and Shift-Invariance , 2006, ICA.

[80]  Jun Zhang,et al.  Divergence Function, Duality, and Convex Analysis , 2004, Neural Computation.

[81]  H. Chernoff.  A Measure of Asymptotic Efficiency for Tests of a Hypothesis Based on the sum of Observations , 1952 .

[82]  Andrzej Cichocki,et al.  On-Line K-PLANE Clustering Learning Algorithm for Sparse Comopnent Analysis , 2006, 2006 IEEE International Conference on Acoustics Speech and Signal Processing Proceedings.

[83]  Pedro A. Valdes-Sosa,et al.  Penalized PARAFAC analysis of spontaneous EEG recordings , 2008 .

[84]  M. Haardt,et al.  Robust methods based on the hosvd for estimating the model order in PARAFAC models , 2008, 2008 5th IEEE Sensor Array and Multichannel Signal Processing Workshop.

[85]  Liqing Zhang,et al.  Flexible Component Analysis for Sparse, Smooth, Nonnegative Coding or Representation , 2007, ICONIP.

[86]  Andrzej Cichocki,et al.  Adaptive blind signal and image processing , 2002 .

[87]  A. Cichocki,et al.  Flexible HALS algorithms for sparse non-negative matrix/tensor factorization , 2008, 2008 IEEE Workshop on Machine Learning for Signal Processing.

[88]  Seungjin Choi,et al.  Algorithms for orthogonal nonnegative matrix factorization , 2008, 2008 IEEE International Joint Conference on Neural Networks (IEEE World Congress on Computational Intelligence).

[89]  Seungjin Choi,et al.  Nonnegative Tucker Decomposition , 2007, 2007 IEEE Conference on Computer Vision and Pattern Recognition.

[90]  NONNEGATIVE RANK FACTORIZATION VIA RANK REDUCTION , 2008 .

[91]  Andrzej Cichocki,et al.  Fast Local Algorithms for Large Scale Nonnegative Matrix and Tensor Factorizations , 2009, IEICE Trans. Fundam. Electron. Commun. Comput. Sci..

[92]  S. Eguchi,et al.  Robust parameter estimation with a small bias against heavy contamination , 2008 .

[93]  K. Bala,et al.  Matrix row-column sampling for the many-light problem , 2007, ACM Trans. Graph..

[94]  Alvaro R. De Pierro,et al.  A modified expectation maximization algorithm for penalized likelihood estimation in emission tomography , 1995, IEEE Trans. Medical Imaging.

[95]  Paris Smaragdis,et al.  Non-negative Matrix Factor Deconvolution; Extraction of Multiple Sound Sources from Monophonic Inputs , 2004, ICA.

[96]  Pando G. Georgiev,et al.  Blind Source Separation Algorithms with Matrix Constraints , 2003, IEICE Trans. Fundam. Electron. Commun. Comput. Sci..

[97]  J. Carroll,et al.  Fitting of the Latent Class model via iteratively reweighted least squares CANDECOMP with nonnegativity constraints , 1989 .

[98]  D J Heeger,et al.  Robust multiresolution alignment of MRI brain volumes , 2000, Magnetic resonance in medicine.

[99]  A. Stegeman,et al.  On Kruskal's uniqueness condition for the Candecomp/Parafac decomposition , 2007 .

[100]  Hyunsoo Kim,et al.  Non-negative Tensor Factorization Based on Alternating Large-scale Non-negativity-constrained Least Squares , 2007, 2007 IEEE 7th International Symposium on BioInformatics and BioEngineering.

[101]  N. Sidiropoulos,et al.  On the uniqueness of multilinear decomposition of N‐way arrays , 2000 .

[102]  Pierre Comon,et al.  Enhanced Line Search: A Novel Method to Accelerate PARAFAC , 2008, SIAM J. Matrix Anal. Appl..

[103]  P ? ? ? ? ? ? ? % ? ? ? ? , 1991 .

[104]  R. Harshman.  The differences between analysis of covariance and correlation , 2001 .

[105]  Christos Boutsidis,et al.  An improved approximation algorithm for the column subset selection problem , 2008, SODA.

[106]  C. R. Rao,et al.  Entropy differential metric, distance and divergence measures in probability spaces: A unified approach , 1982 .

[107]  Lars Kai Hansen,et al.  Parallel Factor Analysis as an exploratory tool for wavelet transformed event-related EEG , 2006, NeuroImage.

[108]  Berkant Savas,et al.  Handwritten digit classification using higher order singular value decomposition , 2007, Pattern Recognit..

[109]  Hong-Ye Gao,et al.  Wavelet Shrinkage Denoising Using the Non-Negative Garrote , 1998 .

[110]  Jacqueline Scherpen,et al.  Proceedings of the 1999 Conference on Information Sciences and Systems , 1999 .

[111]  Andrzej Cichocki,et al.  Nonnegative Tucker decomposition with alpha-divergence , 2008, 2008 IEEE International Conference on Acoustics, Speech and Signal Processing.

[112]  Rasmus Bro,et al.  Multiway analysis of epilepsy tensors , 2007, ISMB/ECCB.

[113]  S. Goreinov,et al.  How to find a good submatrix , 2010 .

[114]  Rasmus Bro,et al.  A comparison of algorithms for fitting the PARAFAC model , 2006, Comput. Stat. Data Anal..

[115]  Mihoko Minami,et al.  Robust Blind Source Separation by Beta Divergence , 2002, Neural Computation.

[116]  Chris H. Q. Ding,et al.  On the Equivalence of Nonnegative Matrix Factorization and Spectral Clustering , 2005, SDM.

[117]  A. R. De Pierro,et al.  On the relation between the ISRA and the EM algorithm for positron emission tomography , 1993, IEEE Trans. Medical Imaging.

[118]  Lieven De Lathauwer,et al.  A Block Component Model-Based Blind DS-CDMA Receiver , 2008, IEEE Transactions on Signal Processing.

[119]  Pierre Comon,et al.  Special Issue on Tensor Decompositions and Applications , 2008, SIAM J. Matrix Anal. Appl..

[120]  Wenwu Wang,et al.  Squared Euclidean Distance Based Convolutive Non-Negative Matrix Factorization with Multiplicative Learning Rules For Audio Pattern Separation , 2007, 2007 IEEE International Symposium on Signal Processing and Information Technology.

[121]  Andrzej Cichocki,et al.  Local Learning Rules for Nonnegative Tucker Decomposition , 2009, ICONIP.

[122]  A. Agresti,et al.  Multiway Data Analysis , 1989 .

[123]  P. Paatero.  A weighted non-negative least squares algorithm for three-way ‘PARAFAC’ factor analysis , 1997 .

[124]  Zhaoshui He,et al.  K-EVD Clustering and Its Applications to Sparse Component Analysis , 2006, ICA.

[125]  Michael David Walsh,et al.  Hong Kong 2001 , 2001 .

[126]  Edgar Velázquez-Armendáriz,et al.  Tensor Clustering for Rendering Many‐Light Animations , 2008 .

[127]  Amnon Shashua,et al.  Nonnegative Sparse PCA , 2006, NIPS.

[128]  Thomas P. Minka,et al.  Divergence measures and message passing , 2005 .

[129]  Romà Tauler,et al.  Chemometrics applied to unravel multicomponent processes and mixtures: Revisiting latest trends in multivariate resolution , 2003 .

[130]  Lucas C. Parra,et al.  Recovery of constituent spectra using non-negative matrix factorization , 2003, SPIE Optics + Photonics.

[131]  Inderjit S. Dhillon,et al.  Matrix Nearness Problems with Bregman Divergences , 2007, SIAM J. Matrix Anal. Appl..

[132]  J Möcks,et al.  Topographic components model for event-related potentials and some biophysical considerations. , 1988, IEEE transactions on bio-medical engineering.

[133]  L. Tucker,et al.  Some mathematical notes on three-mode factor analysis , 1966, Psychometrika.

[134]  Andrzej Cichocki,et al.  Nonnegative matrix factorization with constrained second-order optimization , 2007, Signal Process..

[135]  Pierre Comon,et al.  Nonnegative approximations of nonnegative tensors , 2009, ArXiv.

[136]  Klaas Faber,et al.  Short Communication: On solving generalized eigenvalue problems using Matlab , 1997 .

[137]  H. Kiers,et al.  Selecting among three-mode principal component models of different types and complexities: a numerical convex hull based method. , 2006, The British journal of mathematical and statistical psychology.

[138]  J. Eggert,et al.  Transformation-invariant representation and NMF , 2004, 2004 IEEE International Joint Conference on Neural Networks (IEEE Cat. No.04CH37541).

[139]  Tamara G. Kolda,et al.  Pattern Analysis of Directed Graphs Using DEDICOM: An Application to Enron Email , 2006 .

[140]  Stan Lipovetsky,et al.  PCA and SVD with nonnegative loadings , 2009, Pattern Recognit..

[141]  Dietrich Lehmann,et al.  Nonsmooth nonnegative matrix factorization (nsNMF) , 2006, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[142]  Liviu Badea,et al.  Extracting Gene Expression Profiles Common to Colon and Pancreatic Adenocarcinoma Using Simultaneous Nonnegative Matrix Factorization , 2007, Pacific Symposium on Biocomputing.

[143]  Moody T. Chu,et al.  Low-Dimensional Polytope Approximation and Its Applications to Nonnegative Matrix Factorization , 2008, SIAM J. Sci. Comput..

[144]  Mihoko Minami,et al.  Robust Prewhitening for ICA by Minimizing β-Divergence and Its Application to FastICA , 2007, Neural Processing Letters.

[145]  Tamara G. Kolda,et al.  MATLAB Tensor Toolbox , 2006 .

[146]  Amnon Shashua,et al.  Doubly Stochastic Normalization for Spectral Clustering , 2006, NIPS.

[147]  Michalis Titsias,et al.  Unsupervised learning of multiple objects in images , 2005 .

[148]  J. Kruskal.  Rank, decomposition, and uniqueness for 3-way and n -way arrays , 1989 .

[149]  Bülent Yener,et al.  Unsupervised Multiway Data Analysis: A Literature Survey , 2009, IEEE Transactions on Knowledge and Data Engineering.

[150]  Raul Kompass,et al.  A Generalized Divergence Measure for Nonnegative Matrix Factorization , 2007, Neural Computation.

[151]  Jimeng Sun,et al.  Incremental pattern discovery on streams, graphs and tensors , 2008, SKDD.

[152]  Tamara G. Kolda,et al.  Efficient MATLAB Computations with Sparse and Factored Tensors , 2007, SIAM J. Sci. Comput..

[153]  P R Kennedy,et al.  Direct control of a computer from the human central nervous system. , 2000, IEEE transactions on rehabilitation engineering : a publication of the IEEE Engineering in Medicine and Biology Society.

[154]  A. Ohara.  Geometry of distributions associated with Tsallis statistics and properties of relative entropy minimization , 2007 .

[155]  Patrik O. Hoyer,et al.  Non-negative Matrix Factorization with Sparseness Constraints , 2004, J. Mach. Learn. Res..

[156]  Inderjit S. Dhillon,et al.  Generalized Nonnegative Matrix Approximations with Bregman Divergences , 2005, NIPS.

[157]  Peter D. Turney.  Empirical Evaluation of Four Tensor Decomposition Algorithms , 2007, ArXiv.

[158]  Daniel W. C. Ho,et al.  Underdetermined blind source separation based on sparse representation , 2006, IEEE Transactions on Signal Processing.

[159]  J. Pernier,et al.  Stimulus Specificity of Phase-Locked and Non-Phase-Locked 40 Hz Visual Responses in Human , 1996, The Journal of Neuroscience.

[160]  Chris H. Q. Ding,et al.  Orthogonal nonnegative matrix t-factorizations for clustering , 2006, KDD '06.

[161]  H. Lantéri,et al.  Penalized maximum likelihood image restoration with positivity constraints:multiplicative algorithms , 2002 .

[162]  Wotao Yin,et al.  Iteratively reweighted algorithms for compressive sensing , 2008, 2008 IEEE International Conference on Acoustics, Speech and Signal Processing.

[163]  Imre Csiszár,et al.  Axiomatic Characterizations of Information Measures , 2008, Entropy.

[164]  Sergio Cruces,et al.  Thin QR and SVD factorizations for simultaneous blind signal extraction , 2004, 2004 12th European Signal Processing Conference.

[165]  Charles L. Byrne,et al.  Signal Processing: A Mathematical Approach , 1993 .

[166]  Chih-Jen Lin,et al.  Projected Gradient Methods for Nonnegative Matrix Factorization , 2007, Neural Computation.

[167]  L. De Lathauwer,et al.  Parallel factor analysis by means of simultaneous matrix decompositions , 2005 .

[168]  F. L. Hitchcock.  Multiple Invariants and Generalized Rank of a P‐Way Matrix or Tensor , 1928 .

[169]  Lars Kai Hansen,et al.  Algorithms for Sparse Nonnegative Tucker Decompositions , 2008, Neural Computation.

[170]  Petros Drineas,et al.  Tensor-CUR Decompositions for Tensor-Based Data , 2008, SIAM J. Matrix Anal. Appl..

[171]  M. Daube-Witherspoon,et al.  An Iterative Image Space Reconstruction Algorthm Suitable for Volume ECT , 1986, IEEE Transactions on Medical Imaging.

[172]  H. Sebastian Seung,et al.  Learning the parts of objects by non-negative matrix factorization , 1999, Nature.

[173]  A. Bruckstein,et al.  A Non-Negative and Sparse Enough Solution of an Underdetermined Linear System of Equations is Unique , 2007 .

[174]  I. J. Taneja.  New Developments in Generalized Information Measures , 1995 .

[175]  Timothy R. C. Read,et al.  Goodness-Of-Fit Statistics for Discrete Multivariate Data , 1988 .

[176]  I. Johnstone,et al.  Adapting to Unknown Smoothness via Wavelet Shrinkage , 1995 .

[177]  Inderjit S. Dhillon,et al.  Learning low-rank kernel matrices , 2006, ICML.

[178]  Andrzej Cichocki,et al.  Novel Multi-layer Non-negative Tensor Factorization with Sparsity Constraints , 2007, ICANNGA.

[179]  C. R. Rao,et al.  On the convexity of some divergence measures based on entropy functions , 1982, IEEE Trans. Inf. Theory.

[180]  H. Lantéri,et al.  COMPARISON BETWEEN ISRA AND RLA ALGORITHMS. USE OF A WIENER FILTER BASED STOPPING CRITERION , 1999 .

[181]  P. Kroonenberg.  Applied Multiway Data Analysis , 2008 .

[182]  R. Bro.  Review on Multiway Analysis in Chemistry—2000–2005 , 2006 .

[183]  Joseph F. Murray,et al.  Dictionary Learning Algorithms for Sparse Representation , 2003, Neural Computation.

[184]  Klaus-Robert Müller,et al.  Machine Learning and Applications for Brain-Computer Interfacing , 2007, HCI.

[185]  Yang Li,et al.  Kernel-based multifactor analysis for image synthesis and recognition , 2005, Tenth IEEE International Conference on Computer Vision (ICCV'05) Volume 1.

[186]  Takafumi Kanamori,et al.  Information Geometry of U-Boost and Bregman Divergence , 2004, Neural Computation.

[187]  Seungjin Choi,et al.  A Method of Initialization for Nonnegative Matrix Factorization , 2007, 2007 IEEE International Conference on Acoustics, Speech and Signal Processing - ICASSP '07.

[188]  Chris H. Q. Ding,et al.  Convex and Semi-Nonnegative Matrix Factorizations , 2010, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[189]  Zhaoshui He,et al.  An Efficient K -Hyperplane Clustering Algorithm and Its Application to Sparse Component Analysis , 2007, ISNN.

[190]  J. Chang,et al.  Analysis of individual differences in multidimensional scaling via an n-way generalization of “Eckart-Young” decomposition , 1970 .

[191]  P. Paatero.  Least squares formulation of robust non-negative factor analysis , 1997 .

[192]  A. Rényi.  On Measures of Entropy and Information , 1961 .

[193]  Ananda Sen,et al.  The Theory of Dispersion Models , 1997, Technometrics.

[194]  Andrzej Cichocki,et al.  Fully Online Multicommand Brain-Computer Interface with Visual Neurofeedback Using SSVEP Paradigm , 2007, Comput. Intell. Neurosci..

[195]  A. Cichocki,et al.  Multilayer nonnegative matrix factorisation , 2006 .

[196]  I. Vajda.  Theory of statistical inference and information , 1989 .

[197]  Andrzej Cichocki,et al.  Fast and Efficient Algorithms for Nonnegative Tucker Decomposition , 2008, ISNN.

[198]  Didier G. Leibovici,et al.  Multi-way modelling of high-dimensionality electroencephalographic data , 2001 .

[199]  Andy Harter,et al.  Parameterisation of a stochastic model for human face identification , 1994, Proceedings of 1994 IEEE Workshop on Applications of Computer Vision.

[200]  Tamir Hazan,et al.  Non-negative tensor factorization with applications to statistics and computer vision , 2005, ICML.

[201]  Lieven De Lathauwer,et al.  Swamp reducing technique for tensor decomposition , 2008, 2008 16th European Signal Processing Conference.

[202]  Latent class DEDICOM , 1997 .

[203]  Tamara G. Kolda,et al.  Scalable Tensor Decompositions for Multi-aspect Data Mining , 2008, 2008 Eighth IEEE International Conference on Data Mining.

[204]  H. Jeffreys.  An invariant form for the prior probability in estimation problems , 1946, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[205]  Christoph Schnörr,et al.  Controlling Sparseness in Non-negative Tensor Factorization , 2006, ECCV.

[206]  Motoaki Kawanabe,et al.  Invariant Common Spatial Patterns: Alleviating Nonstationarities in Brain-Computer Interfacing , 2007, NIPS.

[207]  Yonina C. Eldar.  Generalized SURE for Exponential Families: Applications to Regularization , 2008, IEEE Transactions on Signal Processing.

[208]  Shun-ichi Amari,et al.  Methods of information geometry , 2000 .

[209]  Jimeng Sun,et al.  Beyond streams and graphs: dynamic tensor analysis , 2006, KDD '06.

[210]  R. Bro,et al.  A new efficient method for determining the number of components in PARAFAC models , 2003 .

[211]  Andrzej Cichocki,et al.  Neural networks for optimization and signal processing , 1993 .

[212]  P. Hopke,et al.  Application of modified alternating least squares regression to spectroscopic image analysis , 2003 .

[213]  Lucas C. Parra,et al.  Nonnegative matrix factorization for rapid recovery of constituent spectra in magnetic resonance chemical shift imaging of the brain , 2004, IEEE Transactions on Medical Imaging.

[214]  Tamara G. Kolda,et al.  Categories and Subject Descriptors: G.4 [Mathematics of Computing]: Mathematical Software— , 2022 .

[215]  Berkant Savas,et al.  A Newton-Grassmann Method for Computing the Best Multilinear Rank-(r1, r2, r3) Approximation of a Tensor , 2009, SIAM J. Matrix Anal. Appl..

[216]  Shun-ichi Amari,et al.  Differential-geometrical methods in statistics , 1985 .

[217]  P. Green.  Bayesian reconstructions from emission tomography data using a modified EM algorithm. , 1990, IEEE transactions on medical imaging.

[218]  Andrzej Cichocki,et al.  Kernel PCA for Feature Extraction and De-Noising in Nonlinear Regression , 2001, Neural Computing & Applications.

[219]  Mohamed-Jalal Fadili,et al.  Morphological Component Analysis: An Adaptive Thresholding Strategy , 2007, IEEE Transactions on Image Processing.

[220]  L. Lathauwer,et al.  An enhanced plane search scheme for complex-valued tensor decompositions , 2010 .

[221]  Lars Kai Hansen,et al.  ERPWAVELAB A toolbox for multi-channel analysis of time–frequency transformed event related potentials , 2007, Journal of Neuroscience Methods.

[222]  J. Kruskal.  Three-way arrays: rank and uniqueness of trilinear decompositions, with application to arithmetic complexity and statistics , 1977 .

[223]  Gene H. Golub,et al.  Rank-One Approximation to High Order Tensors , 2001, SIAM J. Matrix Anal. Appl..

[224]  Imre Csiszár,et al.  Information Theory - Coding Theorems for Discrete Memoryless Systems, Second Edition , 2011 .