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

TL;DR

This paper introduces a difference-of-convex optimization approach for split feasibility problems involving nonconvex sets, with applications in matrix factorizations and outlier detection, demonstrating convergence and efficiency through numerical experiments.

Contribution

It extends split feasibility problem solutions to nonconvex sets using a difference-of-convex formulation and analyzes convergence of a proximal gradient algorithm.

Findings

Sequences cluster at stationary points under mild conditions

The approach is effective for matrix factorization and outlier detection

Numerical results show efficiency in practical applications

Abstract

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.