Benign landscape for Burer-Monteiro factorizations of MaxCut-type semidefinite programs

TL;DR

This paper establishes a sharp condition on the Laplacian matrix's conditioning that guarantees all second-order critical points of the Burer-Monteiro factorization for MaxCut-type SDPs are global minima, improving understanding of its correctness.

Contribution

The paper provides a precise condition on the Laplacian's conditioning ensuring global optimality of second-order critical points in Burer-Monteiro factorizations for MaxCut SDPs, extending previous results.

Findings

Sharp condition on Laplacian conditioning for global optimality

Improved guarantees for Burer-Monteiro approach in MaxCut SDPs

Application to $ ext{Z}_2$-synchronization and Kuramoto models

Abstract

We consider MaxCut-type semidefinite programs (SDP) which admit a low rank solution. To numerically leverage the low rank hypothesis, a standard algorithmic approach is the Burer-Monteiro factorization, which allows to significantly reduce the dimensionality of the problem at the cost of its convexity. We give a sharp condition on the conditioning of the Laplacian matrix associated with the SDP under which any second-order critical point of the non-convex problem is a global minimizer. By applying our theorem, we improve on recent results about the correctness of the Burer-Monteiro approach on $\mathbb{Z}_2$-synchronization problems and the Kuramoto model.