Consensus in non-commutative spaces

TL;DR

This paper revisits consensus algorithm convergence using Hilbert distance, extending results to non-commutative spaces like positive definite matrices, with applications in quantum stochastic maps.

Contribution

It introduces a novel convergence analysis framework based on Hilbert distance and extends consensus algorithms to non-commutative cones.

Findings

Hilbert distance corresponds to Tsitsiklis Lyapunov function in log coordinates.

Birkhoff theorem guarantees contraction in Hilbert metric for positive homogeneous monotone maps.

Extension of consensus algorithms to positive definite matrices with quantum applications.

Abstract

Convergence analysis of consensus algorithms is revisited in the light of the Hilbert distance. Tsitsiklis Lyapunov function is shown to be the Hilbert distance to consensus in log coordinates. Birkhoff theorem, which proves contraction of the Hilbert metric for any positive homogeneous monotone map, provides an early yet general convergence result for consensus algorithms. Because Birkhoff theorem holds in arbitrary cones, we extend consensus algorithms to the cone of positive definite matrices. The proposed generalization finds applications in the convergence analysis of quantum stochastic maps, which are a generalization of stochastic maps to non-commutative probability spaces.