  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.
