Conic-Optimization Based Algorithms for Nonnegative Matrix Factorization

Abstract

Nonnegative matrix factorization is the following problem: given a nonnegative input matrix $V$ and a factorization rank $K$, compute two nonnegative matrices, $W$ with $K$ columns and $H$ with $K$ rows, such that $WH$ approximates $V$ as well as possible. In this paper, we propose two new approaches for computing high-quality NMF solutions using conic optimization. These approaches rely on the same two steps. First, we reformulate NMF as minimizing a concave function over a product of convex cones--one approach is based on the exponential cone, and the other on the second-order cone. Then, we solve these reformulations iteratively: at each step, we minimize exactly, over the feasible set, a majorization of the objective functions obtained via linearization at the current iterate. Hence these subproblems are convex conic programs and can be solved efficiently using dedicated algorithms. We prove that our approaches reach a stationary point with an accuracy decreasing as $\mathcal{O}(\frac{1}{i})$, where $i$ denotes the iteration number. To the best of our knowledge, our analysis is the first to provide a convergence rate to stationary points for NMF. Furthermore, in the particular cases of rank-one factorizations (that is, $K=1$), we show that one of our formulations can be expressed as a convex optimization problem implying that optimal rank-one approximations can be computed efficiently. Finally, we show on several numerical examples that our approaches are able to frequently compute exact NMFs (that is, with $V = WH$), and compete favorably with the state of the art.