[Paper Review] Around the entropic Talagrand inequality

Giovanni Conforti, Luigia Ripani|arXiv (Cornell University)|Sep 6, 2018

Geometric Analysis and Curvature Flows25 references3 citations

TL;DR

This paper introduces and characterizes entropic Talagrand inequalities, a generalization of classical Talagrand's transportation-entropy inequality where the Wasserstein distance is replaced by an entropic cost derived from Schrödinger bridges. The authors establish equivalent characterizations via reverse hypercontractivity, Hamilton-Jacobi-Bellman semigroup contractivity, and dimension-free concentration, showing that these inequalities imply classical Talagrand and are implied by logarithmic Sobolev inequalities under the Bakry-Émery condition.

ABSTRACT

In this article we study generalization of the classical Talagrand transport-entropy inequality in which the Wasserstein distance is replaced by the entropic transportation cost. This class of inequalities has been introduced in the recent work [9], in connection with the study of Schr\\"odinger bridges. We provide several equivalent characterizations in terms of reverse hypercontractivity for the heat semigroup, contractivity of the Hamilton-Jacobi-Bellman semigroup and dimension-free concentration of measure. Properties such as tensorization and relations to other functional inequalities are also investigated. In particular, we show that the inequalities studied in this article are implied by a Logarithmic Sobolev inequality and imply Talagrand inequality.

Motivation & Objective

• To generalize Talagrand’s transportation-entropy inequality by replacing the Wasserstein distance with an entropic cost derived from the Schrödinger problem.
• To characterize the new class of inequalities through equivalent functional analytic properties such as reverse hypercontractivity and semigroup contractivity.
• To establish that these entropic Talagrand inequalities imply the classical Talagrand inequality and are implied by the logarithmic Sobolev inequality under the Bakry-Émery condition.
• To demonstrate tensorization and provide explicit, concrete conditions for the validity of the inequalities, extending prior abstract results.

Proposed method

• The entropic transportation cost is defined via the Schrödinger problem, minimizing relative entropy over couplings of two measures with respect to the joint law of a Langevin diffusion.
• The authors use the dual formulation of the entropic cost, expressed via the Hamilton-Jacobi-Bellman semigroup, to derive equivalent inequalities involving the semigroup's action on test functions.
• They establish equivalence between the entropic Talagrand inequality and a weak form of reverse hypercontractivity for the heat semigroup associated with the Langevin dynamics.
• The proof strategy leverages known duality results for entropy and relative entropy, including the dual representation of entropy and the additive property of relative entropy.
• The authors apply results from [20] on general transport-entropy inequalities to simplify proofs and extend them to the entropic cost case.
• A key technical tool is the use of the semigroup $ Q_t^ ho $, defined as $ Q_t^ ho φ(x) = -\varepsilon \log P_t^\varepsilon \exp(-\u03c6/\varepsilon)(x) $, which links the entropic cost to the dual formulation of the Schrödinger problem.

Experimental results

Research questions

• RQ1 How can Talagrand’s classical transportation-entropy inequality be generalized using the entropic cost from the Schrödinger problem?
• RQ2 What are the equivalent functional analytic characterizations of the resulting entropic Talagrand inequalities?
• RQ3 How do these inequalities relate to other classical inequalities such as the logarithmic Sobolev and classical Talagrand inequalities?
• RQ4 Under what conditions do these inequalities tensorize, and what does this imply for high-dimensional concentration?
• RQ5 Can explicit conditions be derived for the validity of these inequalities, beyond the abstract framework of general transport costs?

Key findings

• The entropic Talagrand inequality implies the classical Talagrand inequality in the small noise limit $ \varepsilon \to 0 $, confirming consistency with the classical theory.
• The entropic Talagrand inequality is implied by the logarithmic Sobolev inequality under the Bakry-Émery $ \Gamma_2 $ condition, extending Otto-Villani's theorem to the entropic setting.
• The inequality is equivalent to a weak form of reverse hypercontractivity for the semigroup associated with the Langevin dynamics, providing a new functional characterization.
• The entropic Talagrand inequality tensorizes, meaning it holds for product measures under the same condition, which is crucial for high-dimensional applications.
• A new inf-convolution log-Sobolev inequality is derived from the entropic Talagrand inequality, with an explicit constant depending on $ \varepsilon $, $ \lambda $, and $ t $, given by $ 1 + \frac{\varepsilon}{\exp(\lambda \varepsilon t) - (1 + \varepsilon)} $.
• The entropic cost dominates the Wasserstein distance, making the entropic Talagrand inequality strictly stronger than the classical version, and thus provides a tighter concentration bound.