Improved Global Landscape Guarantees for Low-rank Factorization in Synchronization
Shuyang Ling
TL;DR
This work addresses the benign nonconvex landscape of low-rank factorization for orthogonal group synchronization by connecting landscape geometry to the Hessian condition number at the SDR solution. It extends the BM factorization analysis to the Stiefel manifold with $p\ge d+2$ (and $p=d+1$ for small $d$), and provides a master theorem: if $\lambda_{\max}(\boldsymbol{L})/\lambda_{d+1}(\boldsymbol{L})<\alpha_G(p,d)$, then every second-order critical point is global. The authors derive an exact and computable expression for $\alpha_G(p,d)$, along with a simpler bound $\alpha_M(p,d)$, and show substantial improvements over prior results, including the $d=1$ Kuramoto setting and the general $d\ge 2$ cases. They further translate these theoretical gains into practical, high-probability guarantees for SDR-tight regimes and multiple applications such as Kuramoto synchronization, orthogonal group synchronization, and generalized orthogonal Procrustes, highlighting improved noise tolerance and guidance on parameter choices. The methodology relies on a convex relaxation of the condition-number minimization and a novel dual-certificate construction on a broader family of tangent spaces, potentially generalizing to other nonconvex problems with manifold constraints.
Abstract
The orthogonal group synchronization problem, which aims to recover a set of $d \times d$ orthogonal matrices from their pairwise noisy products, plays a fundamental role in signal processing, computer vision, and network analysis. In recent years, numerous optimization techniques, such as semidefinite relaxation (SDR) and low-rank (Burer-Monteiro) factorization, have been proposed to address this problem and their theoretical guarantees have been extensively studied. While SDR is provably powerful and exact in recovering the least-squares estimator under certain mild conditions, it is not scalable. In contrast, the low-rank factorization is highly efficient but nonconvex, meaning its iterates may get trapped in local minima. To close the gap, we analyze the low-rank approach and focus on understanding when the associated nonconvex optimization landscape is benign, i.e., free of spurious local minima. Recent works suggest that the benignness depends on the condition number of the Hessian at the global minimizer, but it remains unclear whether sharp guarantees can be achieved. In this work, we consider the low-rank approach which corresponds to an optimization problem over the Stiefel manifold ${\rm St}(p,d)^{\otimes n}$. By formulating the landscape analysis into another convex optimization problem, we provide a unified characterization of the optimization landscape for all parameter pairs $(p,d)$ with $p \geq d+2$ for $d\geq 1$ and $p = d+1$ for $1\leq d\leq 3$ which gives a much improved dependence on the condition number of the Hessian. Our results recover the known sharp state-of-the-art bound for $d=1$ which is extremely useful for characterizing the Kuramoto synchronization, and significantly improved the guarantees for the general case $d \geq 2$ with $p \geq d+2$ over the existing results. The theoretical results are versatile and applicable to a wide range of examples.