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ENTROPIC CURVATURE ON GRAPHS ALONG SCHRÖDINGER BRIDGES AT ZERO TEMPERATURE

Abstract : Lott-Sturm-Villani theory of curvature on geodesic spaces has been extended to discrete graph spaces by C. Léonard by replacing W 2-Wasserstein geodesics by Schrödinger bridges in the definition of entropic curvature [23, 25, 24]. As a remarkable fact, as a temperature parameter goes to zero, these Schrödinger bridges are supported by geodesics of the space. We analyse this property on discrete graphs to reach entropic curvature on discrete spaces. Our approach provides lower bounds for the entropic curvature for several examples of graphs spaces: the lattice Z n endowed with the counting measure, the discrete cube endowed with product probability measures, the circle, the complete graph, the Bernoulli-Laplace model. As opposed to Erbar-Maas results on graphs [27, 10, 11], entropic curvature results of this paper imply new Prékopa-Leindler type of inequalities on discrete spaces, and new transport-entropy inequalities related to refined concentration properties for the graphs mentioned above. For example on the discrete hypercube {0, 1} n and for the Bernoulli Laplace model, a new W 2 − W 1 transport-entropy inequality is reached, that can not be derived by usual induction argument over the dimension n. As a surprising fact, our method also gives improvements of weak transport-entropy inequalities (see [28, 15]) associated to the so-called convex-hull method by Talagrand [38]. It could also apply to other graphs. The paper starts with a brief overview about known results concerning entropic curvature on discrete graphs. Then it focuses on a particular entropic curvature property on graphs derived from C. Léonard approach [23, 25, 24], and dealing with Schrödinger bridges at zero temperature (see Definition 1.3). The main curvature results are given in section 2, with their connections to new transport-entropy inequalities. The concentration properties following from such transport-entropy inequalities are not developed in the present paper. For that purpose, we refer to [15] by Gozlan & al, where general connections between entropy inequalities and concentration properties are widely investigated. The strategy of proof, presented in section 3, uses the so called slowing-down procedure for Schrödinger bridges associated to jump processes on discrete spaces pushed forward by C. Léonard. General statements , Proposition 3.11 and Lemma 3.1, are obtained by this procedure, which consists of decreasing a temperature parameter γ to 0. The curvature results for the different studied graphs are all derived from Proposition 3.11 and Lemma 3.1. Our strategy also apply for many other graphs which are not considered in this paper. The main goal of this paper is to push forward Leonard's approach for entropic curvature on graphs with few significant results.
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Contributor : Paul-Marie Samson <>
Submitted on : Tuesday, April 21, 2020 - 8:07:35 PM
Last modification on : Friday, April 24, 2020 - 1:47:36 AM

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  • HAL Id : hal-02504530, version 2
  • ARXIV : 2003.05179

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Paul-Marie Samson. ENTROPIC CURVATURE ON GRAPHS ALONG SCHRÖDINGER BRIDGES AT ZERO TEMPERATURE. 2020. ⟨hal-02504530v2⟩

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