Transportation-information Inequalities Research Articles

A characterization of transportation-information inequalities for Markov processes in terms of dimension-free concentration

Il est connu que les inégalités entre les coûts de transport et l’information de Fisher caractérisent certaines propriétés de concentration des processus de Markov autour de leurs mesures invariantes. Cette note apporte une nouvelle caractérisation de l’inégalité quadratique transport-information W2I, en termes d’une propriété de concentration indépendante de la dimension pour les copies i.i.d. (conditionnellement aux positions initiales) du processus de Markov sous-jacent. Ceci est comparable à la caractérisation de Gozlan de l’inégalité quadratique transport-entropie W2H. La preuve est basée sur un nouveau principe de type Laplace pour les normes d’opérateurs des semigroupes de Feynman-Kac, qui peut avoir un intérêt indépendant. Enfin, nous expliquons comment notre théorème et (une forme de) celui de Gozlan sont des exemples d’un principe général de tensorisation convexe-analytique.

The Poincaré inequality and quadratic transportation-variance inequalities

It is known that the Poincare inequality is equivalent to the quadratic transportation-variance inequality (namely $W_{2}^{2}(f\mu ,\mu ) \leqslant C_{V} \mathrm{Var} _{\mu }(f)$), see Jourdain [10] and most recently Ledoux [12]. We give two alternative proofs to this fact. In particular, we achieve a smaller $C_{V}$ than before, which equals the double of Poincare constant. Applying the same argument leads to more characterizations of the Poincare inequality. Our method also yields a by-product as the equivalence between the logarithmic Sobolev inequality and strict contraction of heat flow in Wasserstein space provided that the Bakry-Emery curvature has a lower bound (here the control constants may depend on the curvature bound). Next, we present a comparison inequality between $W_{2}^{2}(f\mu ,\mu )$ and its centralization $W_{2}^{2}(f_{c}\mu ,\mu )$ for $f_{c} = \frac{|\sqrt {f} - \mu (\sqrt {f})|^{2}} {\mathrm{Var} _{\mu }(\sqrt{f} )}$, which may be viewed as some special counterpart of the Rothaus’ lemma for relative entropy. Then it yields some new bound of $W_{2}^{2}(f\mu ,\mu )$ associated to the variance of $\sqrt{f} $ rather than $f$. As a by-product, we have another proof to derive the quadratic transportation-information inequality from Lyapunov condition, avoiding the Bobkov-Gotze’s characterization of the Talagrand’s inequality.

Talagrand concentration inequalities for stochastic partial differential equations

One way to define the concentration of measure phenomenon is via Talagrand inequalities, also called transportation-information inequalities. That is, a comparison of the Wasserstein distance from the given measure to any other absolutely continuous measure with finite relative entropy. Such transportation-information inequalities were recently established for some stochastic differential equations. Here, we develop a similar theory for some stochastic partial differential equations.

Logarithmic Sobolev inequalities for Moebius measures on spheres

Abstract In this paper, using the method in [1], i.e., reduce Moebius measures μ x n {\mu_{x}^{n}} indexed by | x | < 1 {|x|<1} on spheres S n - 1 {S^{n-1}} ( n ≥ 3 {n\geq 3} ) to one-dimensional diffusions on [ 0 , π ] {[0,\pi]} , we obtain that the optimal Poincaré constant is not greater than 2 n - 2 {\frac{2}{n-2}} and the optimal logarithmic Sobolev constant denoted by C LS ⁢ ( μ x n ) {C_{\rm LS}(\mu_{x}^{n})} behaves like 1 n ⁢ log ⁡ ( 1 + 1 1 - | x | ) {\frac{1}{n}\log(1+\frac{1}{1{-}|x|})} . As a consequence, we claim that logarithmic Sobolev inequalities are strictly stronger than L 2 {L^{2}} -transportation-information inequalities.

Wasserstein Continuity of Entropy and Outer Bounds for Interference Channels

It is shown that under suitable regularity conditions, differential entropy is a Lipschitz functional on the space of distributions on $n$-dimensional Euclidean space with respect to the quadratic Wasserstein distance. Under similar conditions, (discrete) Shannon entropy is shown to be Lipschitz continuous in distributions over the product space with respect to Ornstein's $\bar d$-distance (Wasserstein distance corresponding to the Hamming distance). These results together with Talagrand's and Marton's transportation-information inequalities allow one to replace the unknown multi-user interference with its i.i.d. approximations. As an application, a new outer bound for the two-user Gaussian interference channel is proved, which, in particular, settles the "missing corner point" problem of Costa (1985).

A new characterization of quadratic transportation-information inequalities

It is known that a quadratic transportation-information inequality $$\mathrm{W}_2\mathrm{I}$$ interpolates between the Talagrand’s inequality $$\mathrm{W}_2\mathrm{H}$$ and the log-Sobolev inequality (LSI for short). The aim of this paper is threefold: (1) To prove the equivalence of $${\mathrm {W}}_2\mathrm{I}$$ and the Lyapunov condition, which gives a new characterization inspired by Cattiaux et al. (Probab Theory Relat Fields 148(1–2):285–304, 2010). (2) To prove the stability of $${\mathrm {W}}_2\mathrm{I}$$ under bounded perturbations, which gives a transference principle in the sense of Holley–Stroock. (3) To prove $$\mathrm{W}_2\mathrm{H}$$ through a restricted $$\mathrm{W}_2\mathrm{I}$$ , which gives a counterpart of the restricted LSI presented by Gozlan et al. (Ann Probab 39(3):857–880, 2011).

A Link Between the Log-Sobolev Inequality and Lyapunov Condition

We give an alternative look at the log-Sobolev inequality (LSI in short) for log-concave measures by semigroup tools. The similar idea yields a heat flow proof of LSI under some quadratic Lyapunov condition for symmetric diffusions on Riemannian manifolds provided the Bakry-Emery’s curvature is bounded from below. Let’s mention that, the general ϕ-Lyapunov conditions were introduced by Cattiaux et al. (J. Funct. Anal. 256(6), 1821–1841 2009) to study functional inequalities, and the above result on LSI was first proved subject to ϕ(⋅) = d2(⋅,x0) by Cattiaux et al. (Proba. Theory Relat. Fields 148(1–2), 285–304 2010) through a combination of detective L2 transportation-information inequality W2I and the HWI inequality of Otto-Villani. Next, we assert a converse implication that the Lyapunov condition can be derived from LSI, which means their equivalence in the above setting.

Spectral gap, isoperimetry and concentration on trees

We consider the nearest-neighbor model on the finite tree T with generator L. We obtain a twosided estimate of the spectral gap by factor 2. We also identify explicitly the Lipschitzian norm of the operator (–L)−1 in propriate functional space. This leads to the identification of the best constant in the generalized Cheeger isoperimetric inequality on the tree, and to transportation-information inequalities.

Convex concentration for some additive functionals of jump stochastic differential equations

Using forward-backward stochastic calculus, we prove convex concentration inequalities for some additive functionals of the solution of stochastic differential equations with jumps admitting an invariant probability measure. As a consequence, transportation-information inequalities are obtained and bounds on option prices for interest rate derivatives are given as an application.

Equivalence of local transportation-information inequalities and curvature-dimension inequalities

In this short note, we show that the local transportation-information inequalities are equivalent to the in nity curvature-dimension inequalities <italic>CD</italic>(<italic>ρ</italic>,∞) and nite one <italic>CD</italic>(0, <italic>n</italic>) respectively.

Transportation-information inequalities for continuum Gibbs measures

The objective of this paper is to establish explicit concentration inequalities for the Glauber dynamics related with continuum or discrete Gibbs measures. At first we establish the optimal transportation-information $W_1 I$-inequality for the $M/M/\infty$-queue associated with the Poisson measure, which improves several previous known results. Under the Dobrushin's uniqueness condition, we obtain some explicit $W_1 I$-inequalities for Gibbs measures both in the continuum and in the discrete lattice. Our method is a combination of Lipschitzian spectral gap, the Lyapunov test function approach and the tensorization technique.

Bernstein-type Concentration Inequalities for Symmetric Markov Processes

Using the method of transportation-information inequality introduced in [A. Guillin et al., Probab. Theory Related Fields, 144 (2009), pp. 669--695], we establish Bernstein-type concentration inequalities for empirical means of functions of the Markov process $(\int_0^t g(X_s)\,ds)/t$ where $g$ is an unbounded observable of the symmetric Markov process $(X_t)$. Three approaches are proposed: a functional inequalities approach, a Lyapunov function method, and an approach through the Lipschitzian norm of the solution to the Poisson equation. Several applications and examples are studied.

Gradient estimates of Poisson equations on Riemannian manifolds and applications

For the reflected diffusion generated by L = Δ − ∇ V ⋅ ∇ on a connected and complete Riemannian manifold M with empty or convex boundary, we establish some sharp estimates of sup x ∈ M | ∇ G | ( x ) of the Poisson equation − L G = g in terms of the dimension, the diameter and the lower bound of curvature. Applications to transportation-information inequality, to Cheeger's isoperimetric inequality and to Gaussian concentration inequality are given. Several examples are provided.

A note on Talagrand’s transportation inequality and logarithmic Sobolev inequality

We give by simple arguments sufficient conditions, so called Lyapunov conditions, for Talagrand’s transportation information inequality and for the logarithmic Sobolev inequality. Those sufficient conditions work even in the case where the Bakry–Emery curvature is not lower bounded. Several new examples are provided.

Transportation-information inequalities for Markov processes

In this paper, one investigates the transportation-information T c I inequalities: α(T c (ν, μ)) ≤ I (ν|μ) for all probability measures ν on a metric space $${(\mathcal{X}, d)}$$ , where μ is a given probability measure, T c (ν, μ) is the transportation cost from ν to μ with respect to the cost function c(x, y) on $${\mathcal{X}^2}$$ , I(ν|μ) is the Fisher–Donsker–Varadhan information of ν with respect to μ and α : [0, ∞) → [0, ∞] is a left continuous increasing function. Using large deviation techniques, it is shown that T c I is equivalent to some concentration inequality for the occupation measure of a μ-reversible ergodic Markov process related to I(·|μ). The tensorization property of T c I and comparisons of T c I with Poincaré and log-Sobolev inequalities are investigated. Several easy-to-check sufficient conditions are provided for special important cases of T c I and several examples are worked out.

Prescribing a System of Random Variables by Conditional Distributions

Prescribing a System of Random Variables by Conditional Distributions

The Description of a Random Field by Means of Conditional Probabilities and Conditions of Its Regularity

The Description of a Random Field by Means of Conditional Probabilities and Conditions of Its Regularity