The non-convex Burer-Monteiro approach works on smooth semidefinite programs

TL;DR

This paper demonstrates that the low-rank Burer-Monteiro approach reliably finds global optima in certain smooth semidefinite programs, explaining its empirical success across various applications.

Contribution

It provides a theoretical proof that the Burer-Monteiro method almost never has spurious local optima for a broad class of SDPs.

Findings

Low-rank Burer-Monteiro formulation has no spurious local optima in the studied class.

The approach applies to SDPs in max-cut, community detection, robust PCA, phase retrieval, and synchronization.

Empirical observations of convergence are supported by theoretical guarantees.

Abstract

Semidefinite programs (SDPs) can be solved in polynomial time by interior point methods, but scalability can be an issue. To address this shortcoming, over a decade ago, Burer and Monteiro proposed to solve SDPs with few equality constraints via rank-restricted, non-convex surrogates. Remarkably, for some applications, local optimization methods seem to converge to global optima of these non-convex surrogates reliably. Although some theory supports this empirical success, a complete explanation of it remains an open question. In this paper, we consider a class of SDPs which includes applications such as max-cut, community detection in the stochastic block model, robust PCA, phase retrieval and synchronization of rotations. We show that the low-rank Burer--Monteiro formulation of SDPs in that class almost never has any spurious local optima.