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优化算法，用于解决多机器人旋转平均和平移估计问题，这些问题源于协同同时定位与地图构建（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黑塞矩阵与适当加权图的拉普拉斯矩阵之间的理论关系。我们利用这些结果设计了一个协同求解器，在该求解器中，机器人与中央服务器协调，通过每次迭代求解一个拉普拉斯系统来执行近似二阶优化。关键在于，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的机制，并具有可证明的保证。我们对所提方法进行了严格的理论分析，并证明它们具有（局部）线性收敛速度。此外，我们展示了我们的方法可以与渐进非凸性结合，以实现鲁棒性离群值估计。在真实SLAM和SfM场景上的大量实验表明，我们的方法具有优越的收敛速度和通信效率。