Random Pairwise Gossip on Hadamard manifolds

In the context of sensor networks, the consensus problem is an important problem. The goal is to find a distributed algorithm to reach a consensus, i.e. some common value shared by all the agents in the network. Such a well known algorithm is the so-called Random Pairwise Gossip (RPG) [BGPS06]. This algorithm relies on pairwise arithmetic averages. Therefore, it does not apply to some situations of interest where computing arithmetic averages is meaningless. For instance, consensus on axis orientation, or subspace tracking, or camera position, cannot be addressed by RPG. However, all these cases exhibit a common underlying structure: the data belong to some Riemannian manifold [DC92]. The goal of this paper is to adapt the RPG algorithm to the Riemannian manifolds framework. Replacing arithmetic average by midpoints for the metric is a natural idea, promoted in this paper. However, due to curvature, and contrarily to the Euclidean case, convergence is no more guaranteed. However, we show that under suitable curvature assumptions - namely, nonpositive curvature - convergence can still be ensured. Numerical experiments validate our approach.