A Combinatorial Approach to $$L_1$$ L 1 -Matrix Factorization

Abstract

Recent work on low-rank matrix factorization has focused on the missing data problem and robustness to outliers and therefore the problem has often been studied under the $$L_1$$ -norm. However, due to the non-convexity of the problem, most algorithms are sensitive to initialization and tend to get stuck in a local optimum. In this paper, we present a new theoretical framework aimed at achieving optimal solutions to the factorization problem. We define a set of stationary points to the problem that will normally contain the optimal solution. It may be too time-consuming to check all these points, but we demonstrate on several practical applications that even by just computing a random subset of these stationary points, one can achieve significantly better results than current state of the art. In fact, in our experimental results we empirically observe that our competitors rarely find the optimal solution and that our approach is less sensitive to the existence of multiple local minima.