We propose fast and communication-efficient optimization algorithms for multi-robot rotation averaging and translation estimation problems that arise from collaborative simultaneous localization and mapping (SLAM), structure-from-motion (SfM), and camera network localization applications. Our methods are based on theoretical relations between the Hessians of the underlying Riemannian optimization problems and the Laplacians of suitably weighted graphs. We leverage these results to design a collaborative solver in which robots coordinate with a central server to perform approximate second-order optimization, by solving a Laplacian system at each iteration. Crucially, our algorithms permit robots to employ spectral sparsification to sparsify intermediate dense matrices before communication, and hence provide a mechanism to trade off accuracy with communication efficiency with provable guarantees. We perform rigorous theoretical analysis of our methods and prove that they enjoy (local) linear rate of convergence. Furthermore, we show that our methods can be combined with graduated non-convexity to achieve outlier-robust estimation. Extensive experiments on real-world SLAM and SfM scenarios demonstrate the superior convergence rate and communication efficiency of our methods.

翻译：我们提出快速且通信高效的多机器人旋转平均与平移估计算法，这些算法源自协同同时定位与建图（SLAM）、运动恢复结构（SfM）以及相机网络定位应用。我们的方法基于底层黎曼优化问题的海森矩阵与适当加权图拉普拉斯矩阵之间的理论关系。利用这些结果，我们设计了一种协同求解器，其中机器人与中央服务器协作，通过每轮迭代求解一个拉普拉斯系统来执行近似二阶优化。关键之处在于，我们的算法允许机器人在通信前使用谱稀疏化对中间稠密矩阵进行稀疏化，从而提供一种在精度与通信效率之间进行权衡并具有可证明保证的机制。我们对方法进行了严格的理论分析，证明了其具有（局部）线性收敛速度。此外，我们展示了这些方法可以与渐进非凸性相结合以实现鲁棒的离群值抑制。在真实世界的SLAM和SfM场景上的大量实验证明了我们方法优越的收敛速度和通信效率。