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Dobrushin ergodicity coefficient for Markov operators on cones, and beyond

Stéphane Gaubert 1, 2 Zheng Qu 1, 2
2 MAXPLUS - Max-plus algebras and mathematics of decision
Inria Saclay - Ile de France, CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
Abstract : The analysis of classical consensus algorithms relies on contraction properties of adjoints of Markov operators, with respect to Hilbert's pro- jective metric or to a related family of seminorms (Hopf's oscillation or Hilbert's seminorm). We generalize these properties to abstract consensus operators over normal cones, which include the unital completely positive maps (Kraus operators) arising in quantum information theory. In par- ticular, we show that the contraction rate of such operators, with respect to the Hopf oscillation seminorm, is given by an analogue of Dobrushin's ergodicity coefficient. We derive from this result a characterization of the contraction rate of a non-linear flow, with respect to Hopf's oscillation seminorm and to Hilbert's projective metric.
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https://hal.inria.fr/hal-00935284
Contributor : Zheng Qu <>
Submitted on : Thursday, January 23, 2014 - 12:27:00 PM
Last modification on : Thursday, March 5, 2020 - 6:23:48 PM

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  • HAL Id : hal-00935284, version 1

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Stéphane Gaubert, Zheng Qu. Dobrushin ergodicity coefficient for Markov operators on cones, and beyond. International Linear Algebra Society, Jun 2013, Providence, United States. ⟨hal-00935284⟩

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