Sharp, strong and unique minimizers for low complexity robust recovery

TL;DR

This paper investigates the roles of sharp and strong minima in robust recovery, providing verifiable characterizations for convex regularized problems including sparsity and low-rank cases.

Contribution

It offers new quantitative and verifiable characterizations of sharp minima for convex regularized optimization, especially for decomposable norm problems.

Findings

Sharp minima are crucial for robust recovery.

Unique solutions in group-sparsity are strong minima.

Characterizations apply to sparsity, group-sparsity, and low-rank problems.

Abstract

In this paper, we show the important roles of sharp minima and strong minima for robust recovery. We also obtain several characterizations of sharp minima for convex regularized optimization problems. Our characterizations are quantitative and verifiable especially for the case of decomposable norm regularized problems including sparsity, group-sparsity, and low-rank convex problems. For group-sparsity optimization problems, we show that a unique solution is a strong solution and obtain quantitative characterizations for solution uniqueness.