Sharp Restricted Isometry Bounds for the Inexistence of Spurious Local Minima in Nonconvex Matrix Recovery

TL;DR

This paper establishes sharp RIP constant thresholds, specifically $\,rac{1}{2}$ for rank-1 matrix recovery, ensuring no spurious local minima exist and guaranteeing exact recovery from arbitrary initial points.

Contribution

The paper introduces a novel proof technique to determine precise RIP thresholds that prevent spurious local minima in nonconvex matrix recovery.

Findings

RIP constant $\,<1/2$ is necessary and sufficient for exact recovery in rank-1 case.

A local recovery guarantee is provided for initial points with sufficiently low objective value.

The technique sharpens understanding of RIP bounds for nonconvex matrix recovery.

Abstract

Nonconvex matrix recovery is known to contain no spurious local minima under a restricted isometry property (RIP) with a sufficiently small RIP constant $\delta$. If $\delta$ is too large, however, then counterexamples containing spurious local minima are known to exist. In this paper, we introduce a proof technique that is capable of establishing sharp thresholds on $\delta$ to guarantee the inexistence of spurious local minima. Using the technique, we prove that in the case of a rank-1 ground truth, an RIP constant of $\delta<1/2$ is both necessary and sufficient for exact recovery from any arbitrary initial point (such as a random point). We also prove a local recovery result: given an initial point $x_{0}$ satisfying $f(x_{0})\le(1-\delta)^{2}f(0)$, any descent algorithm that converges to second-order optimality guarantees exact recovery.