Average consensus on Riemannian manifolds with bounded curvature

Abstract

Consensus algorithms are a popular choice for computing averages and other similar quantities in ad-hoc wireless networks. However, existing algorithms mostly address the case where the measurements live in a Euclidean space. In this paper, we propose distributed algorithms for averaging measurements lying in a Riemannian manifold. We first propose a direct extension of the classical average consensus algorithm and derive sufficient conditions for its convergence to a consensus configuration. Such conditions depend on the network connectivity, the geometric configuration of the measurements and the curvature of the manifold. However, the consensus configuration to which the algorithm converges may not coincide with the Fréchet mean of the measurements. We thus propose a second algorithm that performs consensus in the tangent space. This algorithm is guaranteed to converge to the Fréchet mean of the measurements, but needs to be initialized at a consensus configuration. By combining these two methods, we obtain a distributed algorithm that converges to the Fréchet mean of the measurements. We test the proposed algorithms on synthetic data sampled from manifolds such as the space of rotations, the sphere and the Grassmann manifold.