Matrix factorization with Binary Components
This addresses a combinatorial optimization problem in computational biology, but it is incremental as it builds on prior work in non-negative matrix factorization.
The paper tackles the problem of low-rank matrix factorization with binary constraints on one factor, motivated by computational biology, and provides an algorithm that provably recovers the exact factorization with O(m r 2^r + mnr + r^2 n) operations.
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 $2^{m \cdot r}$, 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 $O(m r 2^r + mnr + r^2 n)$ operations for $n$ datapoints. To obtain this result, we use theory around the Littlewood-Offord lemma from combinatorics.