Riemannian Consensus for Manifolds With Bounded Curvature

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. DRAFT IEEE TRANSACTIONS ON AUTOMATIC CONTROL 3 conditions for the convergence of the proposed algorithms to a consensus configuration (i.e., where all the nodes converge to the same estimate). We also point out analogies and differences with respect to the Euclidean case.Our work has several important contributions with respect to the state of the art. First, our formulation is completely intrinsic, in the sense that it is not tied to a specific embedding of the manifold. Second, we consider more general (complete and not necessarily compact) Riemannian manifolds. Third, we provide sufficient conditions for the local and, in special cases, global convergence to the sub-manifold of consensus configurations. These conditions depend on the network connectivity, the geometric configuration of the measurements and the curvature of the manifold. We also provide stronger results that hold when additional assumptions on the manifold and network connectivity are made. Finally, we show that, while Euclidean consensus converges to the Euclidean mean of the initial measurements, the Riemannian extension does not converge to the Fréchet mean, which is the Riemannian equivalent of the Euclidean mean.Paper outline. In §II we review Euclidean consensus and relevant notions from Riemannian geometry and optimization. In §III we describe our extension of consensus algorithms to data in manifolds. Our main contributions are presented in §IV and §V. We first give convergence results for the case of general manifolds. We then strengthen our results for the particular case of manifolds with constant, non-negative curvature. In §VI we test the proposed algorithm on manifolds such as the special orthogonal group, the sphere and the Grassmann manifold. In the Appendix we report all the additional derivations and proofs that support the claims stated in the paper. II. MATHEMATICAL BACKGROUNDIn this section, we review some basic concepts related to Euclidean consensus, Riemannian geometry and optimization that are relevant to our development in the rest of the paper. A. Review of Euclidean consensusConsider a network with N nodes. We represent the network as a connected, undirected gr...

The Space of Essential Matrices as a Riemannian Quotient Manifold

“…One of the simplest ones is the sectional curvature K σ(v,w) (x), which denotes the curvature of M at a point x when restricted to a subspace σ(v, w) ⊂ T x M spanned by two linearly independent vectors v, w ∈ T x M. The exact definition of K σ(v,w) (x) is not needed in this paper; however, intuitively one can think of this quantity as a way to measure how fast two geodesics starting at x in the directions u and v either spread (negative curvature) or converge (positive curvature) with respect to similar geodesics in Euclidean space (which has zero curvature). In practice, knowing bounds on the curvature of the space allows to derive convergence guarantees for optimization algorithms such as those described in [2,27] and the Weiszfeld algorithm which we will use in subsection 7.2.…”

Global converegence properties of a consensus protocol on the n-sphere

“…Under certain assumptions regarding the connectivity of G, local consensus on SO(3) can be established with the region of attraction being the largest geodesically convex sets on S n , i.e., open hemispheres. See for example [17] in the case of an undirected graph and [20] in the case of a directed and time-varying graph. A global stability result for discretetime consensus on SO(3) is provided in [22].…”

“…A global stability result for discretetime consensus on SO(3) is provided in [22]. Almost global asymptotical stability of the consensus manifold on the nsphere is known to hold when the graph is a tree [17] or is complete in the case of first-and second-order models [8], [10]. The author of [8] conjectures that global stability also holds for a larger class of topologies whereas [9] provides counter-examples of basic consensus protocols that fail to generate consensus on S 1 .…”

“…The following result, Theorem 8, may be considered as one of the many known facts concerning consensus on convex subsets of manifolds [17]. To solve Problem 4, this paper provides a companion to Theorem 8, Theorem 9 below, that regards all equilibrium manifolds of System 3 under Algorithm 6.…”

“…The main contribution of this paper is to provide a convergence results on a global level for all connected, undirected graph topologies, i.e., for a much larger class of topologies than what has previously been found on the n-sphere [2], [8], [10], [17], [23]. Unlike [22], the control law, which is well-known, does not require the use of a reshaping function depending on the graph topology, i.e., on unavailable information.…”