By Andrzej Cichocki, Rafal Zdunek, Anh Huy Phan, Shun-ichi Amari

This publication presents a wide survey of types and effective algorithms for Nonnegative Matrix Factorization (NMF). This contains NMF’s quite a few extensions and differences, particularly Nonnegative Tensor Factorizations (NTF) and Nonnegative Tucker Decompositions (NTD). NMF/NTF and their extensions are more and more used as instruments in sign and photograph processing, and information research, having garnered curiosity as a result of their power to supply new insights and proper information regarding the complicated latent relationships in experimental info units. it is recommended that NMF supplies significant elements with actual interpretations; for instance, in bioinformatics, NMF and its extensions were effectively utilized to gene expression, series research, the practical characterization of genes, clustering and textual content mining. As such, the authors specialise in the algorithms which are most valuable in perform, the quickest, such a lot powerful, and compatible for large-scale models.Key features:Acts as a unmarried resource reference advisor to NMF, collating info that's commonly dispersed in present literature, together with the authors’ personal lately constructed strategies within the topic area.Uses generalized expense capabilities reminiscent of Bregman, Alpha and Beta divergences, to give sensible implementations of various kinds of powerful algorithms, particularly Multiplicative, Alternating Least Squares, Projected Gradient and Quasi Newton algorithms.Provides a comparative research of the several tools with a purpose to establish approximation mistakes and complexity.Includes pseudo codes and optimized MATLAB resource codes for the majority algorithms awarded within the book.The expanding curiosity in nonnegative matrix and tensor factorizations, in addition to decompositions and sparse illustration of information, will make sure that this publication is vital examining for engineers, scientists, researchers, practitioners and graduate scholars throughout sign and snapshot processing; neuroscience; information mining and information research; machine technological know-how; bioinformatics; speech processing; biomedical engineering; and multimedia.

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**Example text**

BJ ], we can express the squared Euclidean cost function as J(a1 , a1 , . . , aJ , b1 , b2 , . . , bJ ) = = 1 ||Y − ABT ||2F 2 1 ||Y − 2 J aj bTj ||2F . 50b) for j = 1, 2, . . , J subject to a ≥ 0 and b ≥ 0, respectively. 1 Large-scale NMF In many applications, especially in dimension reduction applications the data matrix Y ∈ RI×T can be very large (with millions of entries), but it can be approximately factorized using a rather smaller number of nonnegative components (J), that is, J << I and J << T .

Consider a mapping yt → φ(yt ) or Y → φ(Y) = [φ(y1 ), φ(y2 ), . . , φ(yT )], then the kernel NMF can be defined as φ(Y) ∼ = φ(Y) W BT . 37) which depends only on the kernel K = φT (Y)φ(Y). 12 Convolutive NMF The Convolutive NMF (CNMF) is a natural extension and generalization of the standard NMF. In the Convolutive NMF, we process a set of nonnegative matrices or patterns which are horizontally shifted (or time delayed) versions of the primary matrix X [126]. 38) p=0 where Y ∈ RI×T is a given input data matrix, Ap ∈ RI×J is a set of unknown nonnegative basis matrices, + + p→ 0→ X = X ∈ RJ×T is a matrix representing primary sources or patterns, X is a shifted by p columns version of + p→ X.

P − 1) knowing only the input data matrix Y ∈ RI×T . Each operator Sp = T1 (p = 1, 2, . . , P − 1) performs a horizontal shift of the columns in X by one spot. 13 Overlapping NMF In Convolutive NMF we perform horizontal shift of the columns of the matrix X. In some applications, such as in spectrogram decomposition, we need to perform different transformations by shifting vertically the rows of the matrix X. 39) → ←p Tp is the horizontal-shift matrix operator such that X = X T p and X = X T p .