Abstract

Motivated by an application in computational biology, we consider low-rank ma-trix factorization with {0, 1}-constraints on one of the factors and optionally con-vex constraints on the second one. In addition to the non-convexity shared with other matrix factorization schemes, our problem is further complicated by a com-binatorial constraint set of size 2m·r, wherem 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 + r2n) operations for n datapoints. To obtain this result, we use theory around the Littlewood-Offord lemma from combinatorics. 1