Random pairwise gossip on CAT(κ) metric spaces

Abstract

In the context of sensor networks, gossip algorithms are a popular, well established technique, for achieving consensus when sensor data are encoded in linear spaces. Gossip algorithms also have several extensions to non linear data spaces. Most of these extensions deal with Riemannian manifolds and use Riemannian gradient descent. This paper, instead, studies gossip in a broader CAT(κ) metric setting, encompassing, but not restricted to, several interesting cases of Riemannian manifolds. As it turns out, convergence can be guaranteed as soon as the data lie in a small enough ball of a mere CAT(κ) metric space. We also study convergence speed in this setting and establish linear rates of convergence.