FAST-Sync：任意矩阵李群的快速群同步
FAST-Sync: Fast Group Synchronization for any Matrix Lie Group
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中文总结 AI 辅助
本文提出Fast-Sync，一种针对任意矩阵李群群同步问题的快速线性近似初始化方法，通过利用Kronecker积结构和图拓扑提升速度与精度，使局部优化器在噪声下高效恢复全局最优解。
中文摘要 AI 辅助
群同步（GS）是在给定一组成对比率 $g_i^{-1} g_j$ 的噪声测量值的情况下，估计群 $\mathcal{G}$ 中 $N$ 个未知元素 $g_1,\ldots, g_N \in \mathcal{G}$ 的问题。GS 问题是机器人和计算机视觉中许多状态估计任务的核心，包括 3D 视觉、机器人建图、惯性导航和分子重建。不幸的是，GS 问题通常既是高维的又是非凸的，因此一般难以求解。在本文中，我们提出了 Fast-Sync，一种适用于初始化基于流形的局部优化器或可认证的全局方法的快速线性近似方法。我们的方法将弦初始化推广到任意矩阵李群，并另外提出了两个关键的算法改进：我们展示了如何利用问题数据矩阵中的 Kronecker 积结构以及同步图的拓扑结构来提高速度、可扩展性和准确性。在多个 GS 任务上的实验评估表明，Fast-Sync 提供了高质量的初始化，使局部优化器能够高效地恢复全局最优的 GS 解，即使在相当大的测量噪声下也能实现高成功率。
英文摘要
Group synchronization (GS) is the problem of estimating a set of $N$ unknown elements $g_1,\ldots, g_N \in \mathcal{G}$ in a group $\mathcal{G}$, given noisy measurements of a subset of their pairwise ratios $g_i^{-1} g_j$. GS problems lie at the core of many state estimation tasks in robotics and computer vision, including 3D vision, robotic mapping, inertial navigation, and molecular reconstruction. Unfortunately, GS problems are typically both high-dimensional and non-convex, and therefore hard to solve in general. In this paper, we present Fast-Sync, a fast linear approximation method for GS that is suitable for initializing local manifold-based optimizers or certifiable global methods. Our approach generalizes chordal initialization to arbitrary matrix Lie groups, and additionally proposes two new key algorithmic enhancements: we show how to exploit both the Kronecker-product structure in the problem data matrix and the topology of the synchronization graph to improve speed, scalability, and accuracy. Experimental evaluation across several GS tasks demonstrates that Fast-Sync provides high-quality initializations that enable local optimizers to efficiently recover globally optimal GS solutions, achieving high success rates even with considerable measurement noise.
发表机构
- Northeastern University（东北大学）
- Georgia Institute of Technology（佐治亚理工学院）
机构由 AI 辅助整理，请以论文原文为准。
