A short proof that NMF is NP-hard

We give a short combinatorial proof that the nonnegative matrix factorization is an NP-hard problem. Moreover, we prove that NMF remains NP-hard when restricted to 01-matrices, answering a recent question of Moitra. The (exact) nonnegative matrix factorization is the following problem. Given an integer k and a matrix A with nonnegative entries, do there exist k nonnegative rank-one matrices that sum to A? The smallest k for which this is possible is called the nonnegative rank of A and denoted by rank + (A). We give a short combinatorial proof of a seminal result of Vavasis [6] stating that NMF is NP-hard. Moreover, we prove that NMF remains hard when restricted to Boolean matrices, answering a recent question of Moitra [4].