Consensus under general convexity

Abstract

Abstract-A method is proposed to characterize contraction of a set through orthogonal projections. For discrete-time multi-agent systems, quantitative estimates of convergence (to a consensus) rate are provided by means of contracting convex sets. Required convexity for the sets that should include the values that the transition maps of agents take is considered in a more general sense than that of Euclidean geometry.

Network Consensus in the Wasserstein Metric Space of Probability Measures

“…more generally; see also [43,44]. It is worth noting in passing the related work in [24,28] which deals with similar consensus topics in CAT(0) spaces, and [26] which deals with consensus in a general class of convex metric spaces.…”

“…In particular, this embedding approach is used to perform consensus on the special orthogonal group and on Grassmann manifolds. The authors in [24,26,28] study consensus in different metric spaces which is more closely related to the present work. For example, the author of [28] develops an analogue of Wolfowitz's theorem [3] for a class of metric spaces with non-positive curvature which leads to a notion of consensus in such spaces.…”

“…A survey on synchronisation is given in [22] while consensus and synchronisation are related in [23]. Some other notable exceptions of consensus in non-Euclidean spaces are [24][25][26][27][28][29]. In particular, [25,27] consider general nonlinear consensus on manifolds by embedding such manifolds in a suitable high-dimensional Euclidean space.…”

Network Consensus in the Wasserstein Metric Space of Probability Measures

“…In [30] a consensus algorithm on the Riemannian manifold of (Gaussian) covariance matrices is introduced under the Fisher metric (related to to the Kullback--Leibler divergence). The authors in [25,27,29] study consensus in different metric spaces which is more closely related to the present work. For example, the author of [29] develops an analogue of Wolfowitz's theorem [3] for a class of metric spaces with nonpositive curvature which leads to a notion of consensus in such spaces.…”

“…A survey on synchronization is given in [22,23] while consensus and synchronization are related in [24]. Some other notable exceptions of consensus in non-Euclidean spaces are [25,26,27,28,29,30]. In particular, [26,28] consider general nonlinear consensus on manifolds by embedding such manifolds in a suitable high-dimensional Euclidean space.…”