On the Complexity of Nonnegative Matrix Factorization

Nonnegative matrix factorization (NMF) has become a prominent technique for the analysis of image databases, text databases, and other information retrieval and clustering applications. The problem is most naturally posed as continuous optimization. In this report, we define an exact version of NMF. Then we establish several results about exact NMF: (i) that it is equivalent to a problem in polyhedral combinatorics; (ii) that it is NP-hard; and (iii) that a polynomial-time local search heuristic exists.