Consensus in non-commutative spaces

Convergence analysis of consensus algorithms is revisited in the light of the Hilbert distance. The Lyapunov function used in the early analysis by Tsitsiklis 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.

• Type: Journal
• Year: 2010
• Where: CDC
• Authors: Rodolphe Sepulchre, Alain Sarlette, Pierre Rouchon
• Keywords: CDC 2010, Consensus Algorithms, Control Systems, Convergence Analysis, Hilbert Distance