Abstract

In the Nonnegative Matrix Factorization (NMF) problem we are given an n×m nonnegative matrix M and an integer r> 0. Our goal is to express M as AW where A and W are nonnegative matrices of size n×r and r×m respectively. In some applications, it makes sense to ask instead for the product AW to approximate M – i.e. (approximately) minimize ‖M −AW‖F where ‖‖F denotes the Frobenius norm; we refer to this as Approximate NMF. This problem has a rich history spanning quantum mechanics, probability theory, data anal-ysis, 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