Geometric characterizations for strong minima with applications to nuclear norm minimization problems

TL;DR

This paper introduces geometric characterizations for strong minima in optimization, providing new conditions for nuclear norm minimization and establishing bounds for exact low-rank matrix recovery.

Contribution

It offers weaker geometric conditions than traditional ones, leading to tighter bounds on measurements needed for exact low-rank matrix recovery.

Findings

New geometric characterizations for strong minima

Necessary and sufficient conditions for nuclear norm minimization

Tight bounds on measurements for low-rank matrix recovery

Abstract

In this paper, we introduce several geometric characterizations for strong minima of optimization problems. Applying these results to nuclear norm minimization problems allows us to obtain new necessary and sufficient quantitative conditions for this important property. Our characterizations for strong minima are weaker than the Restricted Injectivity and Nondegenerate Source Condition, which are usually used to identify solution uniqueness of nuclear norm minimization problems. Consequently, we obtain the minimum (tight) bound on the number of measurements for (strong) exact recovery of low-rank matrices.