Sepulchre, R and Sarlette, A and Rouchon, P (2010) *Consensus in non-commutative spaces.* In: UNSPECIFIED pp. 6596-6601..

## 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. ©2010 IEEE.

Item Type: | Conference or Workshop Item (UNSPECIFIED) |
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Subjects: | UNSPECIFIED |

Divisions: | Div F > Control |

Depositing User: | Cron Job |

Date Deposited: | 18 May 2016 17:46 |

Last Modified: | 30 Jun 2016 08:55 |

DOI: | 10.1109/CDC.2010.5717072 |