Characterization of Information Channels for Asymptotic Mean Stationarity and Stochastic Stability of Non-stationary/Unstable Linear Systems

Stabilization of non-stationary linear systems over noisy communication channels is considered. Stochastically stable sources, and unstable but noise-free or bounded-noise systems have been extensively studied in information theory and control theory literature since 1970s, with a renewed interest in the past decade. There have also been studies on non-causal and causal coding of unstable/non-stationary linear Gaussian sources. In this paper, tight necessary and sufficient conditions for stochastic stabilizability of unstable (non-stationary) possibly multi-dimensional linear systems driven by Gaussian noise over discrete channels (possibly with memory and feedback) are presented. Stochastic stability notions include recurrence, asymptotic mean stationarity and sample path ergodicity, and the existence of finite second moments. Our constructive proof uses random-time state-dependent stochastic drift criteria for stabilization of Markov chains. For asymptotic mean stationarity (and thus sample path ergodicity), it is sufficient that the capacity of a channel is (strictly) greater than the sum of the logarithms of the unstable pole magnitudes for memoryless channels and a class of channels with memory. This condition is also necessary under a mild technical condition. Sufficient conditions for the existence of finite average second moments for such systems driven by unbounded noise are provided.Comment: To appear in IEEE Transactions on Information Theor

Quantitative ergodicity for some switched dynamical systems

We provide quantitative bounds for the long time behavior of a class of Piecewise Deterministic Markov Processes with state space Rd \times E where E is a finite set. The continuous component evolves according to a smooth vector field that switches at the jump times of the discrete coordinate. The jump rates may depend on the whole position of the process. Under regularity assumptions on the jump rates and stability conditions for the vector fields we provide explicit exponential upper bounds for the convergence to equilibrium in terms of Wasserstein distances. As an example, we obtain convergence results for a stochastic version of the Morris-Lecar model of neurobiology

On the Geometric Ergodicity of Metropolis-Hastings Algorithms for Lattice Gaussian Sampling

Sampling from the lattice Gaussian distribution is emerging as an important problem in coding and cryptography. In this paper, the classic Metropolis-Hastings (MH) algorithm from Markov chain Monte Carlo (MCMC) methods is adapted for lattice Gaussian sampling. Two MH-based algorithms are proposed, which overcome the restriction suffered by the default Klein's algorithm. The first one, referred to as the independent Metropolis-Hastings-Klein (MHK) algorithm, tries to establish a Markov chain through an independent proposal distribution. We show that the Markov chain arising from the independent MHK algorithm is uniformly ergodic, namely, it converges to the stationary distribution exponentially fast regardless of the initial state. Moreover, the rate of convergence is explicitly calculated in terms of the theta series, leading to a predictable mixing time. In order to further exploit the convergence potential, a symmetric Metropolis-Klein (SMK) algorithm is proposed. It is proven that the Markov chain induced by the SMK algorithm is geometrically ergodic, where a reasonable selection of the initial state is capable to enhance the convergence performance.Comment: Submitted to IEEE Transactions on Information Theor

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