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Consensus in non-commutative spaces

Abstract : 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.
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Contributor : François Chaplais <>
Submitted on : Tuesday, March 15, 2011 - 4:15:06 PM
Last modification on : Thursday, September 24, 2020 - 5:04:18 PM

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Rodolphe Sepulchre, Alain Sarlette, Pierre Rouchon. Consensus in non-commutative spaces. 49th IEEE Conference on Decision and Control, Dec 2010, Atlanta, United States. pp.6596-6601, ⟨10.1109/CDC.2010.5717072⟩. ⟨hal-00576914⟩

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