Benign landscape for Burer-Monteiro factorizations of MaxCut-type semidefinite programs
Faniriana Rakoto Endor, Irène Waldspurger
TL;DR
This work establishes a sharp, deterministic condition on the Laplacian conditioning of a MaxCut-type SDP under Burer-Monteiro factorization that guarantees the landscape is benign, i.e., every second-order critical point is globally optimal. The core idea is to bound the condition number λ_n(L)/λ_2(L) by the factor p, using a dual-certificate construction that proves all nonoptimal second-order points cannot occur when p exceeds this ratio. The main theorem yields improved, near-optimal recovery guarantees for Z2-synchronization under Gaussian and Bernoulli noise and for the Kuramoto model, linking spectral properties of the problem to reliable nonconvex optimization. The approach combines explicit gradient/Hessian expressions with a variational duality framework to certify optimality via a carefully chosen dual certificate, and it tightens previous results in the literature. Overall, the paper provides a deterministic criterion that explains and extends the empirical success of Burer-Monteiro methods for low-rank MaxCut-type SDPs in important signal-processing and synchronization problems.
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.