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

This e-book offers a extensive survey of versions and effective algorithms for Nonnegative Matrix Factorization (NMF). This comprises NMF’s numerous extensions and adjustments, in particular Nonnegative Tensor Factorizations (NTF) and Nonnegative Tucker Decompositions (NTD). NMF/NTF and their extensions are more and more used as instruments in sign and snapshot processing, and knowledge research, having garnered curiosity because of their power to supply new insights and suitable information regarding the advanced 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 useful characterization of genes, clustering and textual content mining. As such, the authors specialise in the algorithms which are most respected in perform, the quickest, so much powerful, and appropriate for large-scale types.

**Key positive factors: **

- Acts as a unmarried resource reference consultant to NMF, collating info that's extensively dispersed in present literature, together with the authors’ personal lately built innovations within the topic sector.
- Uses generalized fee capabilities comparable to Bregman, Alpha and Beta divergences, to provide sensible implementations of various kinds of strong algorithms, particularly Multiplicative, Alternating Least Squares, Projected Gradient and Quasi Newton algorithms.
- Provides a comparative research of different tools so one can establish approximation errors and complexity.
- Includes pseudo codes and optimized MATLAB resource codes for the majority algorithms offered within the publication.

The expanding curiosity in nonnegative matrix and tensor factorizations, in addition to decompositions and sparse illustration of knowledge, will make sure that this publication is vital analyzing for engineers, scientists, researchers, practitioners and graduate scholars throughout sign and photo processing; neuroscience; facts mining and information research; machine technology; bioinformatics; speech processing; biomedical engineering; and multimedia.

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**Extra info for Nonnegative matrix and tensor factorizations**

**Example text**

33) and we can apply the transpose operator to give YT = XT WT YT + ET . 34) general, the convex NMF applies to both nonnegative data and mixed sign data which can be written symbolically as Y± = Y± W+ X+ + E. 9(b)). The convex NMF usually implies that both nonnegative factors A and B = XT tend to be very sparse. 35) j=1 where λj is the positive j-th eigenvalue (a diagonal entry of diagonal matrix ) and vj is the corresponding J eigenvector for the eigenvalue decomposition: YT Y = V VT = j=1 λj vj vTj .

Such a model is often referred to as Nonnegative Factorization (NF) [58,59]. 13) where nonnegativity constraints are imposed to all or only to the selected factor matrices: A ∈ RI×J , S ∈ RJ×R , and/or X ∈ RR×T . It should be noted that if we do not impose any additional constraints to the factors (besides nonnegativity), the three-factor NMF can be reduced to the standard (two-factor) NMF by the transformation A ← AS or X ← SX. However, the three-factor NMF is not equivalent to the standard NMF if we apply special constraints or conditions as illustrated by the following special cases.

For = 0, S = IJ , the model reduces to the standard NMF and for → 1 strong smoothing is imposed on S, causing increased sparseness of both A and X in order to maintain the faithfulness of the model. , lowpass filtering to perform smoothing or highpass filtering in order to remove slowly changing trends from the estimated components (source signals). 15) where F is a suitably designed (prescribed) filtering matrix. In the case of lowpass filtering, we usually perform some kind of averaging in the sense that every sample value xjt is replaced by a weighted average of that value and the neighboring value, so that in the simplest scenario the smoothing lowpass filtering matrix F can take the following form: ⎡ 1/2 1/3 0 0 ⎤ ⎢ 1/2 1/3 1/3 0⎥ ⎢ ⎥ ⎢ ⎥ 1/3 1/3 1/3 ⎢ ⎥ ⎢ ⎥ ∈ RT ×T .