# Dobrushin ergodicity coefficient for Markov operators on cones, and beyond

arXiv: https://arxiv.org/abs/1302.5226 · HTML: https://mlsanity.com/paper/1302.5226/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 Feb 2013.

## Built on by (most cited first, 1 of 1)

- [Support Topology and Gradient Mixing in Sinkhorn Layers](https://mlsanity.com/paper/2609.07954/) (2026)

## Abstract (arXiv)

The analysis of classical consensus algorithms relies on contraction properties of adjoints of Markov operators, with respect to Hilbert's projective 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 particular, 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.