Improved Global Landscape Guarantees for Low-rank Factorization in Synchronization

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.