By Andrzej Cichocki, Rafal Zdunek, Anh Huy Phan, Shun-ichi Amari
This booklet offers a vast survey of versions and effective algorithms for Nonnegative Matrix Factorization (NMF). This comprises NMF’s a number of extensions and transformations, 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 information research, having garnered curiosity because of their potential to supply new insights and correct information regarding the complicated latent relationships in experimental information units. it's endorsed that NMF grants 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 specialize in the algorithms which are most respected in perform, taking a look at the quickest, such a lot strong, and appropriate for large-scale versions.
Key positive aspects:
- Acts as a unmarried resource reference consultant to NMF, collating info that's broadly dispersed in present literature, together with the authors’ personal lately built innovations within the topic sector.
- Uses generalized price features akin to Bregman, Alpha and Beta divergences, to offer functional implementations of different types of strong algorithms, specifically Multiplicative, Alternating Least Squares, Projected Gradient and Quasi Newton algorithms.
- Provides a comparative research of different tools that allows you to establish approximation blunders and complexity.
- Includes pseudo codes and optimized MATLAB resource codes for the majority algorithms provided within the publication.
The expanding curiosity in nonnegative matrix and tensor factorizations, in addition to decompositions and sparse illustration of information, will make sure that this ebook is vital studying for engineers, scientists, researchers, practitioners and graduate scholars throughout sign and photograph processing; neuroscience; facts mining and knowledge research; computing device technology; bioinformatics; speech processing; biomedical engineering; and multimedia.
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Additional resources for Nonnegative matrix and tensor factorizations
16) 0 1/3 1/2 A standard way of performing highpass filtering is equivalent to an application of a first-order differential operator, which means (in the simplest scenario) just replacing each sample value by the difference between 12 Nonnegative Matrix and Tensor Factorizations the value at that point and the value at the preceding point. For example, a highpass filtering matrix can take following form (using the first order or second order discrete difference forms): ⎡ 1 −1 0 0 ⎤ ⎢ −1 2 −1 0⎥ ⎢ ⎥ ⎢ ⎥ −1 2 −1 ⎢ ⎥ ⎥ ∈ RT ×T .
10) subject to nonnegativity constraints A ≥ 0 and X ≥ 0 (component-wise) and an additional orthogonality constraint: AT A = IJ or XXT = IJ . Probably the simplest and most efficient way to impose orthogonality onto the matrix A or X is to perform the following transformation after each iteration A ← A AT A −1/2 , or X ← XXT −1/2 X. 5 Semi-NMF and Nonnegative Factorization of Arbitrary Matrix In some applications the observed input data are unsigned (unconstrained or bipolar) as indicated by Y = Y± ∈ RI×T which allows us to relax the constraints regarding nonnegativity of one factor (or only specific vectors of a matrix).
17). Two special subarrays have more compact representations: the j-th column of matrix A, a: j , may also be denoted as aj , whereas the q-th frontal slice of a third-order tensor, Y: : q may also be denoted as Yq , (q = 1, 2, . . , Q). 3 Unfolding – Matricization It is often very convenient to represent tensors as matrices or to represent multi-way relationships and a tensor decomposition in their matrix forms. Unfolding, also known as matricization or flattening, is a process of reordering the elements of an N-th order tensor into a matrix.