Provable Burer-Monteiro factorization for a class of norm-constrained matrix problems

TL;DR

This paper analyzes the convergence of a non-convex projected gradient descent method, ProjFGD, for low-rank matrix problems with norm constraints, demonstrating its effectiveness in quantum tomography and phase retrieval.

Contribution

It introduces ProjFGD, a novel algorithm with theoretical guarantees for non-convex low-rank matrix optimization under constraints, extending recent unconstrained results.

Findings

Non-convex projected gradient descent achieves local linear convergence.

ProjFGD outperforms existing methods in quantum state tomography.

Theoretical extension of descent lemma to constrained problems.

Abstract

We study the projected gradient descent method on low-rank matrix problems with a strongly convex objective. We use the Burer-Monteiro factorization approach to implicitly enforce low-rankness; such factorization introduces non-convexity in the objective. We focus on constraint sets that include both positive semi-definite (PSD) constraints and specific matrix norm-constraints. Such criteria appear in quantum state tomography and phase retrieval applications. We show that non-convex projected gradient descent favors local linear convergence in the factored space. We build our theory on a novel descent lemma, that non-trivially extends recent results on the unconstrained problem. The resulting algorithm is Projected Factored Gradient Descent, abbreviated as ProjFGD, and shows superior performance compared to state of the art on quantum state tomography and sparse phase retrieval applications.