The orthogonal group synchronization problem, which focuses on recovering orthogonal group elements from their corrupted pairwise measurements, encompasses examples such as high-dimensional Kuramoto model on general signed networks, $\mathbb{Z}_2$-synchronization, community detection under stochastic block models, and orthogonal Procrustes problem. The semidefinite relaxation (SDR) has proven its power in solving this problem; however, its expensive computational costs impede its widespread practical applications. We consider the Burer-Monteiro factorization approach to the orthogonal group synchronization, an effective and scalable low-rank factorization to solve large scale SDPs. Despite the significant empirical successes of this factorization approach, it is still a challenging task to understand when the nonconvex optimization landscape is benign, i.e., the optimization landscape possesses only one local minimizer, which is also global. In this work, we demonstrate that if the degree of freedom within the factorization exceeds twice the condition number of the ``Laplacian" (certificate matrix) at the global minimizer, the optimization landscape is absent of spurious local minima. Our main theorem is purely algebraic and versatile, and it seamlessly applies to all the aforementioned examples: the nonconvex landscape remains benign under almost identical condition that enables the success of the SDR. Additionally, we illustrate that the Burer-Monteiro factorization is robust to ``monotone adversaries", mirroring the resilience of the SDR. In other words, introducing ``favorable" adversaries into the data will not result in the emergence of new spurious local minimizers.

翻译：正交群同步问题旨在从受扰动的成对测量中恢复正交群元素，其涵盖的实例包括一般符号网络上的高维仓本模型、$\mathbb{Z}_2$-同步、随机块模型下的社区检测以及正交普鲁克问题。半定规划松弛（SDR）已被证明能有效求解该问题，但其高昂的计算成本阻碍了广泛的实践应用。我们采用Burer-Monteiro分解方法处理正交群同步问题，这是一种可扩展的低秩分解策略，用于求解大规模半定规划。尽管该分解方法在实证中取得了显著成功，但理解非凸优化景观何时呈现良性特征——即优化景观仅存在一个同时也是全局最优解的局部极小点——仍具挑战性。本研究表明，若分解内部的自由度超过全局极小点处“拉普拉斯”矩阵（验证矩阵）条件数的两倍，则优化景观将不存在伪局部极小点。我们的主要定理是纯代数且普适的，可无缝应用于所有前述实例：在保证SDR成功的几乎相同条件下，非凸景观始终保持良性。此外，我们证明Burer-Monteiro分解对“单调扰动”具有鲁棒性，这与SDR的稳健特性相呼应。换言之，在数据中引入“有利”扰动不会导致新的伪局部极小点产生。