Consensus on Lie groups for the Riemannian Center of Mass

TL;DR

This paper introduces a novel, efficient distributed algorithm for computing the Riemannian center of mass on Lie groups, leveraging Riemannian gradient flow and gradient tracking, with proven convergence and superior performance.

Contribution

It develops the first intrinsic, distributed algorithm for RCM on Lie groups using Riemannian gradient flow and gradient tracking, with global convergence guarantees.

Findings

Proposed algorithm converges to the RCM point under certain conditions.

The method demonstrates faster convergence compared to existing algorithms.

The approach is applicable to distributed systems on Lie groups.

Abstract

In this paper, we develop a consensus algorithm for distributed computation of the Riemannian center of mass (RCM) on Lie Groups. The algorithm is built upon a distributed optimization reformulation that allows developing an intrinsic, distributed (without relying on a consensus subroutine), and a computationally efficient protocol for the RCM computation. The novel idea for developing this fast distributed algorithm is to utilize a Riemannian version of distributed gradient flow combined with a gradient tracking technique. We first guarantee that, under certain conditions, the limit point of our algorithm is the RCM point of interest. We then provide a proof of global convergence in the Euclidean setting, that can be viewed as a "geometric" dynamic consensus that converges to the average from arbitrary initial points. Finally, we proceed to showcase the superior convergence properties of the proposed approach as compared with other classes of consensus optimization-based algorithms for the RCM computation.