Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userCharacterization of a class of weak transport-entropy inequalities on the line
160
0
0.0
(
0
)
Authors
Nathael Gozlan
Ask ChatGPT about the research
No Arabic abstract
We study an optimal weak transport cost related to the notion of convex order between probability measures. On the real line, we show that this weak transport cost is reached for a coupling that does not depend on the underlying cost function. As an application, we give a necessary and sufficient condition for weak transport-entropy inequalities in dimension one. In particular, we obtain a weak transport-entropy form of the convex Poincar{e} inequality in dimension one.
rate research
Read More
We introduce the notion of an interpolating path on the set of probability measures on finite graphs. Using this notion, we first prove a displacement convexity property of entropy along such a path and derive Prekopa-Leindler type inequalities, a Talagrand transport-entropy inequality, certain HWI type as well as log-Sobolev type inequalities in discrete settings. To illustrate through examples, we apply our results to the complete graph and to the hypercube for which our results are optimal -- by passing to the limit, we recover the classical log-Sobolev inequality for the standard Gaussian measure with the optimal constant.
Firstly, we derive in dimension one a new covariance inequality of $L_{1}-L_{infty}$ type that characterizes the isoperimetric constant as the best constant achieving the inequality. Secondly, we generalize our result to $L_{p}-L_{q}$ bounds for the covariance. Consequently, we recover Cheegers inequality without using the co-area formula. We also prove a generalized weighted Hardy type inequality that is needed to derive our covariance inequalities and that is of independent interest. Finally, we explore some consequences of our covariance inequalities for $L_{p}$-Poincar{e} inequalities and moment bounds. In particular, we obtain optimal constants in general $L_{p}$-Poincar{e} inequalities for measures with finite isoperimetric constant, thus generalizing in dimension one Cheegers inequality, which is a $L_{p}$-Poincar{e} inequality for $p=2$, to any real $pgeq 1$.
We investigate the role of convexity in Renyi entropy power inequalities. After proving that a general Renyi entropy power inequality in the style of Bobkov-Chistyakov (2015) fails when the Renyi parameter $rin(0,1)$, we show that random vectors with $s$-concave densities do satisfy such a Renyi entropy power inequality. Along the way, we establish the convergence in the Central Limit Theorem for Renyi entropies of order $rin(0,1)$ for log-concave densities and for compactly supported, spherically symmetric and unimodal densities, complementing a celebrated result of Barron (1986). Additionally, we give an entropic characterization of the class of $s$-concave densities, which extends a classical result of Cover and Zhang (1994).
Mixtures are convex combinations of laws. Despite this simple definition, a mixture can be far more subtle than its mixed components. For instance, mixing Gaussian laws may produce a potential with multiple deep wells. We study in the present work fine properties of mixtures with respect to concentration of measure and Sobolev type functional inequalities. We provide sharp Laplace bounds for Lipschitz functions in the case of generic mixtures, involving a transportation cost diameter of the mixed family. Additionally, our analysis of Sobolev type inequalities for two-component mixtures reveals natural relations with some kind of band isoperimetry and support constrained interpolation via mass transportation. We show that the Poincare constant of a two-component mixture may remain bounded as the mixture proportion goes to 0 or 1 while the logarithmic Sobolev constant may surprisingly blow up. This counter-intuitive result is not reducible to support disconnections, and appears as a reminiscence of the variance-entropy comparison on the two-point space. As far as mixtures are concerned, the logarithmic Sobolev inequality is less stable than the Poincare inequality and the sub-Gaussian concentration for Lipschitz functions. We illustrate our results on a gallery of concrete two-component mixtures. This work leads to many open questions.
This paper contributes to the study of class $(Sigma^{r})$ as well as the c`adl`ag semi-martingales of class $(Sigma)$, whose finite variational part is c`adl`ag instead of continuous. The two above-mentioned classes of stochastic processes are extensions of the family of c`adl`ag semi-martingales of class $(Sigma)$ considered by Nikeghbali cite{nik} and Cheridito et al. cite{pat}; i.e., they are processes of the class $(Sigma)$, whose finite variational part is continuous. The two main contributions of this paper are as follows. First, we present a new characterization result for the stochastic processes of class $(Sigma^{r})$. More precisely, we extend a known characterization result that Nikeghbali established for the non-negative sub-martingales of class $(Sigma)$, whose finite variational part is continuous (see Theorem 2.4 of cite{nik}). Second, we provide a framework for unifying the studies of classes $(Sigma)$ and $(Sigma^{r})$. More precisely, we define and study a new larger class that we call class $(Sigma^{g})$. In particular, we establish two characterization results for the stochastic processes of the said class. The first one characterizes all the elements of class $(Sigma^{g})$. Hence, we derive two corollaries based on this result, which provides new ways to characterize classes $(Sigma)$ and $(Sigma^{r})$. The second characterization result is, at the same time, an extension of the above mentioned characterization result for class $(Sigma^{r})$ and of a known characterization result of class $(Sigma)$ (see Theorem 2 of cite{fjo}). In addition, we explore and extend the general properties obtained for classes $(Sigma)$ and $(Sigma^{r})$ in cite{nik,pat,mult, Akdim}.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
