Riemannian Consensus for Manifolds with Bounded Curvature

TL;DR

This paper introduces a Riemannian consensus algorithm that extends averaging consensus to manifolds with bounded curvature, enabling distributed computation of averages on complex geometric spaces.

Contribution

It presents an intrinsic Riemannian consensus algorithm applicable to any complete Riemannian manifold, with convergence analysis and practical testing on various manifolds.

Findings

Algorithm converges under certain curvature conditions.

Effective on manifolds like rotations, sphere, Grassmann.

Outperforms Euclidean methods on manifold data.

Abstract

Consensus algorithms are popular distributed algorithms for computing aggregate quantities, such as averages, in ad-hoc wireless networks. However, existing algorithms mostly address the case where the measurements lie in a Euclidean space. In this work we propose Riemannian consensus, a natural extension of the existing averaging consensus algorithm to the case of Riemannian manifolds. Unlike previous generalizations, our algorithm is intrinsic and, in principle, can be applied to any complete Riemannian manifold. We characterize our algorithm by giving sufficient convergence conditions on Riemannian manifolds with bounded curvature and we analyze the differences that rise with respect to the classical Euclidean case. We test the proposed algorithms on synthetic data sampled from manifolds such as the space of rotations, the sphere and the Grassmann manifold.