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Generalized Logarithmic Sobolev Inequality by the JKO Scheme

Thibault Caillet, Fanch Coudreuse

TL;DR

This work addresses generalized logarithmic Sobolev inequalities for log-concave measures $\\eta=e^{-V}$ by blending Bakry–Émery analysis with optimal transport through a discrete JKO scheme. It develops two dissipation identities—entropy dissipation via geodesic convexity and Fisher information dissipation via the five-gradients inequality—along a discrete flow and proves convergence to the equilibrium, yielding modified LSIs that interpolate between gradient-flow and transport-based methods. The main contributions include a general theorem under moduli of convexity and monotonicity, recovery of classical LSIs when $V$ is strongly convex, and simpler sufficient conditions plus a radial corollary; the framework also clarifies how to derive various non-quadratic LSIs within a unified discrete-flow approach. The significance lies in providing a unifying, discretized gradient-flow strategy that connects Bakry–Émery and optimal transport techniques, with potential to extend to other functional inequalities and broader potential landscapes.

Abstract

Using a discrete Bakry-{É}mery method based on the JKO scheme, relying on the dissipation of entropy and Fisher information along a discrete flow, we establish new generalized logarithmic Sobolev inequality for log-concave measures of the form $e^{-V} under strict convexity assumptions on $V$ . We then show how this method recovers some well-known inequalities. This approach can be viewed as interpolating between the Bakry-{É}mery method and optimal transport techniques based on geodesic convexity.

Generalized Logarithmic Sobolev Inequality by the JKO Scheme

TL;DR

This work addresses generalized logarithmic Sobolev inequalities for log-concave measures by blending Bakry–Émery analysis with optimal transport through a discrete JKO scheme. It develops two dissipation identities—entropy dissipation via geodesic convexity and Fisher information dissipation via the five-gradients inequality—along a discrete flow and proves convergence to the equilibrium, yielding modified LSIs that interpolate between gradient-flow and transport-based methods. The main contributions include a general theorem under moduli of convexity and monotonicity, recovery of classical LSIs when is strongly convex, and simpler sufficient conditions plus a radial corollary; the framework also clarifies how to derive various non-quadratic LSIs within a unified discrete-flow approach. The significance lies in providing a unifying, discretized gradient-flow strategy that connects Bakry–Émery and optimal transport techniques, with potential to extend to other functional inequalities and broader potential landscapes.

Abstract

Using a discrete Bakry-{É}mery method based on the JKO scheme, relying on the dissipation of entropy and Fisher information along a discrete flow, we establish new generalized logarithmic Sobolev inequality for log-concave measures of the form V$ . We then show how this method recovers some well-known inequalities. This approach can be viewed as interpolating between the Bakry-{É}mery method and optimal transport techniques based on geodesic convexity.
Paper Structure (21 sections, 16 theorems, 65 equations)

This paper contains 21 sections, 16 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. Statement of the Main Result
  3. A Remark on the choice of Entropy and on regularity hypotheses
  4. Comparison with the Literature
  5. The classical Bakry-Emery's Proof
  6. Acknowledgment
  7. Preliminaries
  8. Optimal transportation
  9. Functionals over probabilities
  10. One-Step JKO Scheme
  11. Modulus of convexity and monotonicity
  12. Proof of theorem \ref{['thm: log_Sobolev']}
  13. One-Step dissipation of entropy
  14. One-Step dissipation of Fisher's information
  15. Iteration along the flow
  16. ...and 6 more sections

Key Result

Theorem 1.1

Assume that $V$ admits moduli of convexity $\sigma$ and modulus of monotonicity $\omega$ (see definition def: modulus) on $\Omega$ eventually unbounded. Under the point-wise bound for all $z \neq 0$: Then, for $G := H + L$, $\eta$ satisfies the modified Log-Sobolev inequality: for all $g > 0$ with $\int_\Omega g \dd{\eta} = 1$, and $\nabla \log g \in W^{1,\infty}(\Omega)$ it holds that

Theorems & Definitions (39)

  • Theorem 1.1
  • Definition 2.1: Optimal Transport Problem
  • Theorem 2.2: Kantorovitch Duality and generalized Brenier's theorem
  • Theorem 2.3: Five gradients inequality
  • Definition 2.4: Entropy and Potential Energy
  • Definition 2.5: Generalized Fisher's information
  • Remark 2.6
  • Theorem 2.7
  • proof
  • Definition 2.8: Modulus of monotonicity and convexity
  • ...and 29 more