Dobrushin's ergodicity coefficient for Markov operators on cones

We give a characterization of the contraction ratio of bounded linear maps in Banach space with respect to Hopf's oscillation seminorm, which is the infinitesimal distance associated to Hilbert's projective metric, in terms of the extreme points of a certain abstract "simplex". The formula is then applied to abstract Markov operators defined on arbitrary cones, which extend the row stochastic matrices acting on the standard positive cone and the completely positive unital maps acting on the cone of positive semidefinite matrices. When applying our characterization to a stochastic matrix, we recover the formula of Dobrushin's ergodicity coefficient. When applying our result to a completely positive unital map, we therefore obtain a noncommutative version of Dobrushin's ergodicity coefficient, which gives the contraction ratio of the map (representing a quantum channel or a "noncommutative Markov chain") with respect to the diameter of the spectrum. The contraction ratio of the dual operator (Kraus map) with respect to the total variation distance will be shown to be given by the same coefficient. We derive from the noncommutative Dobrushin's ergodicity coefficient an algebraic characterization of the convergence of a noncommutative consensus system or equivalently the ergodicity of a noncommutative Markov chain.Comment: An announcement of some of the present results has appeared in the Proceedings of the ECC 2013 conference (Zurich). Further results can be found in the companion arXiv:1302.522

Dobrushin ergodicity coefficient for Markov operators on cones, and beyond

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

Bundle-based pruning in the max-plus curse of dimensionality free method

Recently a new class of techniques termed the max-plus curse of dimensionality-free methods have been developed to solve nonlinear optimal control problems. In these methods the discretization in state space is avoided by using a max-plus basis expansion of the value function. This requires storing only the coefficients of the basis functions used for representation. However, the number of basis functions grows exponentially with respect to the number of time steps of propagation to the time horizon of the control problem. This so called "curse of complexity" can be managed by applying a pruning procedure which selects the subset of basis functions that contribute most to the approximation of the value function. The pruning procedures described thus far in the literature rely on the solution of a sequence of high dimensional optimization problems which can become computationally expensive. In this paper we show that if the max-plus basis functions are linear and the region of interest in state space is convex, the pruning problem can be efficiently solved by the bundle method. This approach combining the bundle method and semidefinite formulations is applied to the quantum gate synthesis problem, in which the state space is the special unitary group (which is non-convex). This is based on the observation that the convexification of the unitary group leads to an exact relaxation. The results are studied and validated via examples

Curse of dimensionality reduction in max-plus based approximation methods: theoretical estimates and improved pruning algorithms

Max-plus based methods have been recently developed to approximate the value function of possibly high dimensional optimal control problems. A critical step of these methods consists in approximating a function by a supremum of a small number of functions (max-plus "basis functions") taken from a prescribed dictionary. We study several variants of this approximation problem, which we show to be continuous versions of the facility location and $k$-center combinatorial optimization problems, in which the connection costs arise from a Bregman distance. We give theoretical error estimates, quantifying the number of basis functions needed to reach a prescribed accuracy. We derive from our approach a refinement of the curse of dimensionality free method introduced previously by McEneaney, with a higher accuracy for a comparable computational cost.Comment: 8pages 5 figure