正交群子群上同步问题的统一方法
A Unified Approach to Synchronization Problems over Subgroups of the Orthogonal Group
AI总结：
针对正交群闭子群上的同步问题，提出含初始化和广义幂法迭代的统一非凸方法，给出理论保证与几何条件验证，并通过实验证明其在速度、可扩展性和估计误差上更优。
摘要：
群$\u003cspan style="font-style: italic;"\u003eG\u003c/span\u003e$上的同步问题旨在基于对形如$G^*_i {G^*_j}^{-1}$的所有成对比值的子集的含噪观测，估计一组群元素$G^*_1, \u2026, G^*_n \u2208 \u003cspan style="font-style: italic;"\u003eG\u003c/span\u003e$。该问题近年来受到广泛关注，在诸多科学与工程领域都有应用。本文研究群为正交群闭子群的一类同步问题，这类问题涵盖了实际中出现的许多群同步问题。我们的贡献分为五个方面：第一，提出了求解这类群同步问题的统一方法，包含合适的初始化步骤和基于广义幂法的迭代细化步骤，并证明在关于群、测量图、噪声和初始化的特定假设下，该方法在估计误差上具有强理论保证；第二，提出了本方法所需的两个几何条件，并证明它们对多种实际相关的正交群子群均成立，这些条件与子群的误差界几何——优化中的一个重要概念——密切相关；第三，针对标准随机图和随机矩阵模型，验证了关于测量图和噪声的假设；第四，基于度量熵的经典概念，提出并分析了一种新型谱型估计器；第五，通过大量数值实验表明，我们提出的非凸方法在计算速度、可扩展性和/或估计误差方面优于现有方法。
英文摘要：
The problem of synchronization over a group $\mathcal{G}$ aims to estimate a collection of group elements $G^*_1, \dots, G^*_n \in \mathcal{G}$ based on noisy observations of a subset of all pairwise ratios of the form $G^*_i {G^*_j}^{-1}$. Such a problem has gained much attention recently and finds many applications across a wide range of scientific and engineering areas. In this paper, we consider the class of synchronization problems in which the group is a closed subgroup of the orthogonal group. This class covers many group synchronization problems that arise in practice. Our contribution is fivefold. First, we propose a unified approach for solving this class of group synchronization problems, which consists of a suitable initialization step and an iterative refinement step based on the generalized power method, and show that it enjoys a strong theoretical guarantee on the estimation error under certain assumptions on the group, measurement graph, noise, and initialization. Second, we formulate two geometric conditions that are required by our approach and show that they hold for various practically relevant subgroups of the orthogonal group. The conditions are closely related to the error-bound geometry of the subgroup -- an important notion in optimization. Third, we verify the assumptions on the measurement graph and noise for standard random graph and random matrix models. Fourth, based on the classic notion of metric entropy, we develop and analyze a novel spectral-type estimator. Finally, we show via extensive numerical experiments that our proposed non-convex approach outperforms existing approaches in terms of computational speed, scalability, and/or estimation error.
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