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Conference Papers Year : 2013

Dobrushin ergodicity coefficient for Markov operators on cones, and beyond

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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Dates and versions

hal-00935284 , version 1 (23-01-2014)

Identifiers

  • HAL Id : hal-00935284 , version 1

Cite

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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