Matrix factorization with binary components

Motivated by an application in computational biology, we consider low-rank matrix factorization with {0, 1}-constraints on one of the factors and optionally convex constraints on the second one. In addition to the non-convexity shared with other matrix factorization schemes, our problem is further complicated by a combinatorial constraint set of size 2m·r, where m is the dimension of the data points and r the rank of the factorization. Despite apparent intractability, we provide − in the line of recent work on non-negative matrix factorization by Arora et al. (2012)− an algorithm that provably recovers the underlying factorization in the exact case with O(mr2r +mnr + rn) operations for n datapoints. To obtain this result, we use theory around the Littlewood-Offord lemma from combinatorics.