Projected subgradient methods for paraconvex optimization: Application to robust low-rank matrix recovery

TL;DR

This paper introduces fundamental properties of paraconvex functions, analyzes the convergence of projected subgradient methods with various step-sizes, and demonstrates their effectiveness in robust low-rank matrix recovery tasks.

Contribution

It provides a comprehensive convergence analysis of subgradient methods for paraconvex functions and applies these methods to several robust low-rank matrix recovery problems.

Findings

Convergence of subgradient methods is established under various step-size rules.

Numerical experiments show promising results in robust matrix recovery tasks.

Theoretical analysis extends error bounds to paraconvex functions.

Abstract

This paper is devoted to the class of paraconvex functions and presents some of its fundamental properties, characterization, and examples that can be used for their recognition and optimization. Next, the convergence analysis of the projected subgradient methods with several step-sizes (i.e., constant, nonsummable, square-summable but not summable, geometrically decaying, and Scaled Polyak's step-sizes) to global minima for this class of functions is studied. In particular, the convergence rate of the proposed methods is investigated under paraconvexity and the H\"olderian error bound condition, where the latter is an extension of the classical error bound condition. The preliminary numerical experiments on several robust low-rank matrix recovery problems (i.e., robust matrix completion, image inpainting, robust nonnegative matrix factorization, robust matrix compression, and robust image deblurring) indicate promising behavior for these projected subgradient methods, validating our theoretical foundations.