Benign landscapes for synchronization on spheres via normalized Laplacian matrices

Abstract

We study the nonconvex optimization landscapes of synchronization problems on spheres. First, we present new results for the statistical problem of synchronization over the two-element group $\mathbf{Z}_2$. We consider the nonconvex least-squares problem with $\mathbf{Z}_2 = \{\pm 1\}$ relaxed to the unit sphere in $\mathbf{R}^r$ for $r \geq 2$; for several popular models, including graph clustering under the binary stochastic block model, we show that, for any $r \geq 2$, every second-order critical point recovers the ground truth in the asymptotic regimes where exact recovery is information-theoretically possible. Such statistical optimality via spherical relaxations had previously only been shown for (potentially arbitrarily) larger relaxation dimension $r$. Second, we consider the global synchronization of networks of coupled oscillators under the (homogeneous) Kuramoto model. We prove new and optimal asymptotic results for random signed networks on an Erd\H{o}s--R\'enyi graph, and we give new and simple proofs for several existing state-of-the-art results. Our key tool is a deterministic landscape condition that extends a recent result of Rakoto Endor and Waldspurger. This result says that, if a certain problem-dependent Laplacian matrix has small enough condition number, the nonconvex landscape is benign. Our extension allows the condition number to include an arbitrary diagonal preconditioner, which gives tighter results for many problems. We show that, for the synchronization of Kuramoto oscillator networks on nearest-neighbor circulant graphs as studied by Wiley, Strogatz, and Girvan, this condition is optimal. We also prove a natural complex extension that may be of interest for synchronization on the special orthogonal group $\operatorname{SO}(2)$.