arXiv subjects

Katalin Marton

Publications and source records attributed to Katalin Marton.

5 recordsLinked to original sources

arXiv2015

Logarithmic Sobolev inequalities in discrete product spaces: a proof by a transportation cost distance

The aim of this paper is to prove an inequality between relative entropy and the sum of average conditional relative entropies of the following form: For a fixed probability measure $q^n$ on $\mathcal X^n$, ($\mathcal X$ is a finite set), and any probability measure $p^n=\mathcal L(Y^n)$ on $\mathcal X^n$, we have \begin{equation}\label{*} D(p^n||q^n)\leq Const. \sum_{i=1}^n \Bbb E_{p^n} D(p_i(\cdot|Y_1,\dots, Y_{i-1},Y_{i+1},\dots, Y_n) || q_i(\cdot|Y_1,\dots, Y_{i-1},Y_{i+1},\dots, Y_n)), \end{equation} where $p_i(\cdot|y_1,\dots, y_{i-1},y_{i+1},\dots, y_n)$ and $q_i(\cdot|x_1,\dots, x_{i-1},x_{i+1},\dots, x_n)$ denote the local specifications for $p^n$ resp. $q^n$. The constant shall depend on the properties of the local specifications of $q^n$. Inequality (*) is meaningful in product spaces, both in the discrete and the continuous case, and can be used to prove a logarithmic Sobolev inequality for $q^n$, provided uniform logarithmic Sobolev inequalities are available for $q_i(\cdot|x_1,\dots, x_{i-1},x_{i+1},\dots, x_n)$, for all fixed $i$ and all fixed $(x_1,\dots, x_{i-1},x_{i+1},\dots, x_n)$. Inequality (*) directly implies that the Gibbs sampler associated with $q^n$ is a contraction for relative entropy. We derive inequality (*), and thereby a logarithmic Sobolev inequality, in discrete product spaces, by proving inequalities for an appropriate Wasserstein-like distance. A logarithmic Sobolev inequality is, roughly speaking, a contractivity property of relative entropy with respect to some Markov semigroup. It is much easier to prove contractivity for a distance between measures than for relative entropy, since distances satisfy the triangle inequality, and for them well known linear tools, like estimates through matrix norms can be applied.

math.PR↗

arXiv2015

Bounding relative entropy by the relative entropy of local specifications in product spaces

For a class of density functions $q^n(x^n)$ on $\Bbb R^n$ we prove an inequality between relative entropy and the sum of average conditional relative entropies of the following form: For any density function $p^n(x^n)$ on $\Bbb R^n$, $D(p^n||q^n)\leq Const. \sum_{i=1}^n \Bbb E D(p_i(\cdot|Y_1,..., Y_{i-1},Y_{i+1},..., Y_n) || Q_i(\cdot|Y_1,..., Y_{i-1},Y_{i+1},..., Y_n)),$ where $p_i(\cdot|y_1,..., y_{i-1},y_{i+1},..., y_n)$ and $Q_i(\cdot|x_1,..., x_{i-1},x_{i+1},..., x_n)$ denote the local specifications for $p^n$ resp. $q^n$, i.e., the conditional density functions of the $i$'th coordinate, given the other coordinates. The constant depends on the properties of the local specifications of $q^n$. The above inequality implies a logarithmic Sobolev inequality for $q^n$. We get an explicit lower bound for the logarithmic Sobolev constant of $q^n$ under the assumptions that: (i) the local specifications of $q^n$ satisfy logarithmic Sobolev inequalities with constants $ρ_i$, and (ii) they also satisfy some condition expressing that the mixed partial derivatives of the Hamiltonian of $q^n$ are not too large relative to the logarithmic Sobolev constants $ρ_i$. Condition (ii) may be weaker than that used in Otto and Reznikoff's recent paper on the estimation of logarithmic Sobolev constants of spin systems.

math.PR↗

arXiv2015

An explicit bound on the Logarithmic Sobolev constant of weakly dependent random variables

We prove logarithmic Sobolev inequality for measures $$ q^n(x^n)=\text{dist}(X^n)=\exp\bigl(-V(x^n)\bigr), \quad x^n\in \Bbb R^n, $$ under the assumptions that: (i) the conditional distributions $$ Q_i(\cdot| x_j, j\neq i)=\text{dist}(X_i| X_j= x_j, j\neq i) $$ satisfy a logarithmic Sobolev inequality with a common constant $ρ$, and (ii) they also satisfy some condition expressing that the mixed partial derivatives of the Hamiltonian $V$ are not too large relative to $ρ$. \bigskip Condition (ii) has the form that the norms of some matrices defined in terms of the mixed partial derivatives of $V$ do not exceed $1/2\cdotρ\cdot(1-\de)$. The logarithmic Sobolev constant of $q^n$ can then be estimated from below by $1/2\cdotρ\cdotδ$. This improves on earlier results by Th. Bodineau and B. Helffer, by giving an explicit bound, for the logarithmic Sobolev constant for $q^n$.

math.PR↗

arXiv2012

An inequality for relative entropy and logarithmic Sobolev inequalities in Euclidean spaces

Let $q(x)$ and $p(x)$ denote density functions on the $n$-dimensional Euclidean space, and let $p_i(\cdot|y_1,..., y_{i-1},y_{i+1},..., y_n)$ and $Q_i(\cdot|x_1,..., x_{i-1},x_{i+1},..., x_n)$ denote their local specifications. For a class of density functions $q$ we prove an inequality between the relative entropy $D(p||q)$ and a weighted sum of the conditional relative entropies $D(p_i(\cdot|Y_1,..., Y_{i-1},Y_{i+1},..., Y_n) ||Q_i(\cdot|Y_1,..., Y_{i-1},Y_{i+1},..., Y_n))$ that holds for any $p$. The weights are proportional to the logarithmic Sobolev constants of the local specifications of $q$. Thereby we derive a logarithmic Sobolev inequality for a weighted Gibbs sampler governed by the local specifications of $q$. Moreover, this inequality implies a classical logarithmic Sobolev inequality for $q$, as defined for Gaussian distribution by L. Gross. This strengthens a result by F. Otto and M. Reznikoff. The proof is based on ideas developed by F. Otto and C. Villani in their paper on the connection between Talagrand's transportation-cost inequality and logarithmic Sobolev inequality.

math.FA↗

arXiv2004

Measure concentration for Euclidean distance in the case of dependent random variables

Let q^n be a continuous density function in n-dimensional Euclidean space. We think of q^n as the density function of some random sequence X^n with values in \BbbR^n. For I\subset[1,n], let X_I denote the collection of coordinates X_i, i\in I, and let \bar X_I denote the collection of coordinates X_i, i\notin I. We denote by Q_I(x_I|\bar x_I) the joint conditional density function of X_I, given \bar X_I. We prove measure concentration for q^n in the case when, for an appropriate class of sets I, (i) the conditional densities Q_I(x_I|\bar x_I), as functions of x_I, uniformly satisfy a logarithmic Sobolev inequality and (ii) these conditional densities also satisfy a contractivity condition related to Dobrushin and Shlosman's strong mixing condition.

math.PR↗