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Consensus in non-commutative spaces
Sepulchre, Rodolphe; Sarlette, Alain; Rouchon, Pierre
2010In Proceedings of the 49th IEEE Conference on Decision and Control
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Keywords :
Hilbert metric; positive definite cone; Kraus maps
Abstract :
[en] 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.
Research center :
Systems and Modeling
Disciplines :
Electrical & electronics engineering
Author, co-author :
Sepulchre, Rodolphe ;  Université de Liège - ULiège > Dép. d'électric., électron. et informat. (Inst.Montefiore) > Systèmes et modélisation
Sarlette, Alain ;  Université de Liège - ULiège > Dép. d'électric., électron. et informat. (Inst.Montefiore) > Systèmes et modélisation
Rouchon, Pierre;  Mines ParisTech > Centre Automatique et Systemes > Professor
Language :
English
Title :
Consensus in non-commutative spaces
Publication date :
December 2010
Event name :
49th IEEE Conference on Decision and Control
Event organizer :
IEEE
Event place :
Atlanta, United States - Georgia
Event date :
from 15-12-2010 to 17-12-2010
Main work title :
Proceedings of the 49th IEEE Conference on Decision and Control
Publisher :
IEEE, Atlanta, United States - Georgia
Pages :
6596-6601
Peer reviewed :
Peer reviewed
Funders :
F.R.S.-FNRS - Fonds de la Recherche Scientifique [BE]
Pole d'attraction interuniversitaire DYSCO
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