A difference-of-convex approach for split feasibility with applications to matrix factorizations and outlier detection

The split feasibility problem is to find an element in the intersection of a closed set $C$ and the linear preimage of another closed set $D$, assuming the projections onto $C$ and $D$ are easy to compute. This class of problems arises naturally in many contemporary applications such as compressed sensing. While the sets $C$ and $D$ are typically assumed to be convex in the literature, in this paper, we allow both sets to be possibly nonconvex. We observe that, in this setting, the split feasibility problem can be formulated as an optimization problem with a difference-of-convex objective so that standard majorization-minimization type algorithms can be applied. Here we focus on the nonmonotone proximal gradient algorithm with majorization studied in [15, Appendix A]. We show that, when this algorithm is applied to a split feasibility problem, the sequence generated clusters at a stationary point of the problem under mild assumptions. We also study local convergence property of the sequence under suitable assumptions on the closed sets involved. Finally, we perform numerical experiments to illustrate the efficiency of our approach on solving split feasibility problems that arise in completely positive matrix factorization, (uniformly) sparse matrix factorization, and outlier detection.
Comment: Published in JOGO. In this ArXiv version, 1. We fix an unfortunate error in Lemma 4 and the paragraph after it: the matrices involved should have rank r instead of being full rank; 2. An additional compactness condition is needed for the uniform KL exponent in Proposition 5