Asymptotic Subspace Consensus in Dynamic Networks

TL;DR

This paper studies the problem of asymptotic subspace consensus in dynamic networks, providing a complete characterization of its solvability and analyzing the convergence behavior under various network conditions.

Contribution

It introduces the concept of asymptotic subspace consensus, characterizes its solvability in oblivious message adversaries, and extends existing algorithms to this new setting.

Findings

Characterization of solvability conditions for asymptotic subspace consensus

Algorithms for asymptotic consensus extend to subspace consensus under weaker network assumptions

Bounds on the convergence rate to lower-dimensional subspaces

Abstract

We introduce the problem of asymptotic subspace consensus, which requires the outputs of processes to converge onto a common subspace while remaining inside the convex hull of initial vectors.This is a relaxation of asymptotic consensus in which outputs have to converge to a single point, i.e., a zero-dimensional affine subspace. We give a complete characterization of the solvability of asymptotic subspace consensus in oblivious message adversaries. In particular, we show that a large class of algorithms used for asymptotic consensus gracefully degrades to asymptotic subspace consensus in distributed systems with weaker assumptions on the communication network. We also present bounds on the rate by which a lower-than-initial dimension is reached.