  We present the first distributed optimization algorithm with lazy
communication for collaborative geometric estimation, the backbone of modern
collaborative simultaneous localization and mapping (SLAM) and
structure-from-motion (SfM) applications. Our method allows agents to
cooperatively reconstruct a shared geometric model on a central server by
fusing individual observations, but without the need to transmit potentially
sensitive information about the agents themselves (such as their locations).
Furthermore, to alleviate the burden of communication during iterative
optimization, we design a set of communication triggering conditions that
enable agents to selectively upload a targeted subset of local information that
is useful to global optimization. Our approach thus achieves significant
communication reduction with minimal impact on optimization performance. As our
main theoretical contribution, we prove that our method converges to
first-order critical points with a global sublinear convergence rate. Numerical
evaluations on bundle adjustment problems from collaborative SLAM and SfM
datasets show that our method performs competitively against existing
distributed techniques, while achieving up to 78% total communication
reduction.
