Over-parametrization via Lifting for Low-rank Matrix Sensing: Conversion of Spurious Solutions to Strict Saddle Points

TL;DR

This paper demonstrates that over-parametrization in low-rank matrix sensing transforms spurious stationary points into strict saddle points, facilitating their escape via local search, and provides bounds on the necessary degree of over-parametrization.

Contribution

It introduces an infinite hierarchy of non-convex problems via lifting and Burer-Monteiro factorization, showing over-parametrization converts spurious solutions into strict saddle points.

Findings

Spurious solutions become strict saddle points under over-parametrization.

Over-parametrization creates negative curvature to escape spurious solutions.

Derived bounds on the degree of over-parametrization needed to eliminate spurious solutions.

Abstract

This paper studies the role of over-parametrization in solving non-convex optimization problems. The focus is on the important class of low-rank matrix sensing, where we propose an infinite hierarchy of non-convex problems via the lifting technique and the Burer-Monteiro factorization. This contrasts with the existing over-parametrization technique where the search rank is limited by the dimension of the matrix and it does not allow a rich over-parametrization of an arbitrary degree. We show that although the spurious solutions of the problem remain stationary points through the hierarchy, they will be transformed into strict saddle points (under some technical conditions) and can be escaped via local search methods. This is the first result in the literature showing that over-parametrization creates a negative curvature for escaping spurious solutions. We also derive a bound on how much over-parametrization is requited to enable the elimination of spurious solutions.