NDLI: Computing a Nonnegative Matrix Factorization---Provably

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Arora, Sanjeev Kannan, Ravi Ge, Rong Moitra, Ankur
Copyright Year	2016
Abstract	In the nonnegative matrix factorization (NMF) problem we are given an $n \times m$ nonnegative matrix $M$ and an integer $r > 0$. Our goal is to express $M$ as $A W$, where $A$ and $W$ are nonnegative matrices of size $n \times r$ and $r \times m$, respectively. In some applications, it makes sense to ask instead for the product $AW$ to approximate $M$, i.e. (approximately) minimize $\left\lVert{M - AW}_F\right\rVert$, where $\left\lVert\right\rVert_F$, denotes the Frobenius norm; we refer to this as approximate NMF. This problem has a rich history spanning quantum mechanics, probability theory, data analysis, polyhedral combinatorics, communication complexity, demography, chemometrics, etc. In the past decade NMF has become enormously popular in machine learning, where $A$ and $W$ are computed using a variety of local search heuristics. Vavasis recently proved that this problem is NP-complete. (Without the restriction that $A$ and $W$ be nonnegative, both the exact and approximate problems can be solved optimally via the singular value decomposition.) We initiate a study of when this problem is solvable in polynomial time. Our results are the following: 1. We give a polynomial-time algorithm for exact and approximate NMF for every constant $r$. Indeed NMF is most interesting in applications precisely when $r$ is small. 2. We complement this with a hardness result, that if exact $NMF$ can be solved in time $(nm)^{o(r)}$, 3-SAT has a subexponential-time algorithm. This rules out substantial improvements to the above algorithm. 3. We give an algorithm that runs in time polynomial in $n$, $m$, and $r$ under the separablity condition identified by Donoho and Stodden in 2003. The algorithm may be practical since it is simple and noise tolerant (under benign assumptions). Separability is believed to hold in many practical settings. To the best of our knowledge, this last result is the first example of a polynomial-time algorithm that provably works under a non-trivial condition on the input and we believe that this will be an interesting and important direction for future work.
Starting Page	1582
Ending Page	1611
Page Count	30
File Format	PDF
ISSN	00975397
DOI	10.1137/130913869
e-ISSN	10957111
Journal	SIAM Journal on Computing (SMJCAT)
Issue Number	4 (Special Section on the Forty-Fourth Annual ACM Symposium on Theory of Computing (STOC 2012))
Volume Number	45
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2016-08-30
Access Restriction	Subscribed
Subject Keyword	Analysis of algorithms and problem complexity separability provable algorithms Analysis of algorithms nonnegative matrix factorization
Content Type	Text
Resource Type	Article
Subject	Mathematics Computer Science

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