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arXiv:1401.5226stat.ML

The Why and How of Nonnegative Matrix Factorization

Nicolas Gillis

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Abstract

Nonnegative matrix factorization (NMF) has become a widely used tool for the analysis of high-dimensional data as it automatically extracts sparse and meaningful features from a set of nonnegative data vectors. We first illustrate this property of NMF on three applications, in image processing, text mining and hyperspectral imaging --this is the why. Then we address the problem of solving NMF, which is NP-hard in general. We review some standard NMF algorithms, and also present a recent subclass of NMF problems, referred to as near-separable NMF, that can be solved efficiently (that is, in polynomial time), even in the presence of noise --this is the how. Finally, we briefly describe some problems in mathematics and computer science closely related to NMF via the nonnegative rank.

arXiv comments
25 pages, 5 figures. Some typos and errors corrected, Section 3.2 reorganized
Journal reference
DOI
Subjects
stat.MLMachine Learning (Statistics) cs.IRInformation Retrieval cs.LGMachine Learning math.OCOptimization and Control

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Linked by 1 identifiable commenter in 1 comment across 1 thread, from Feb 2019 to Feb 2019. Found in 1 HN submission from Mar 2019 to Mar 2019. Open submission discussions.

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Discussed in comments1 thread

  1. Nonnegative matrix factorization 1 comment

    • Interesting. The problem of finding a low-rank-approximation (in Frobenius norm) to a given matrix is well understood and solved in reasonable time (truncated SVD, as stated in this article). However, the particular characteristic of this NMF approach seems to be that we require…

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