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Rényi--Sobolev Inequalities and Connections to Spectral Graph Theory

Lei Yu, Hao Wu

TL;DR

This work introduces Rényi--Sobolev inequalities as nonlinear, dimension-free extensions of log-Sobolev inequalities by replacing entropy with a two-parameter entropy Ent_{p,q} tied to Rényi divergences. The authors prove sharp asymptotics: as the dimension n grows, Xi_{p,q}^{(n)}(α) converges to conv Xi_q(α) when p≤q and to 0 when p>q, revealing a parameter-driven transition. The proofs combine information-theoretic characterizations with the method of types, and the results are then connected to contractive/data-processing inequalities, concentration of measure (via generalized Herbst arguments), and spectral graph theory through generalized Rayleigh q-quotients and Faber–Krahn-type problems. They further provide explicit binary-case expressions and specialized corollaries for Gaussian and hypercube settings, illustrating the sharpness and applicability of the framework. The work thus links nonlinear entropy-based inequalities to a broad spectrum of analytic and combinatorial topics, with potential implications for spectral bounds and concentration phenomena in high dimensions.

Abstract

In this paper, we generalize the log-Sobolev inequalities to Rényi--Sobolev inequalities by replacing the entropy with the two-parameter entropy, which is a generalized version of entropy and closely related to Rényi divergences. We derive the sharp nonlinear dimension-free version of this kind of inequalities. Interestingly, the resultant inequalities show a transition phenomenon depending on the parameters. We then connect Rényi--Sobolev inequalities to contractive and data-processing inequalities, concentration inequalities, and spectral graph theory. Our proofs in this paper are based on the information-theoretic characterization of the Rényi--Sobolev inequalities, as well as the method of types.

Rényi--Sobolev Inequalities and Connections to Spectral Graph Theory

TL;DR

This work introduces Rényi--Sobolev inequalities as nonlinear, dimension-free extensions of log-Sobolev inequalities by replacing entropy with a two-parameter entropy Ent_{p,q} tied to Rényi divergences. The authors prove sharp asymptotics: as the dimension n grows, Xi_{p,q}^{(n)}(α) converges to conv Xi_q(α) when p≤q and to 0 when p>q, revealing a parameter-driven transition. The proofs combine information-theoretic characterizations with the method of types, and the results are then connected to contractive/data-processing inequalities, concentration of measure (via generalized Herbst arguments), and spectral graph theory through generalized Rayleigh q-quotients and Faber–Krahn-type problems. They further provide explicit binary-case expressions and specialized corollaries for Gaussian and hypercube settings, illustrating the sharpness and applicability of the framework. The work thus links nonlinear entropy-based inequalities to a broad spectrum of analytic and combinatorial topics, with potential implications for spectral bounds and concentration phenomena in high dimensions.

Abstract

In this paper, we generalize the log-Sobolev inequalities to Rényi--Sobolev inequalities by replacing the entropy with the two-parameter entropy, which is a generalized version of entropy and closely related to Rényi divergences. We derive the sharp nonlinear dimension-free version of this kind of inequalities. Interestingly, the resultant inequalities show a transition phenomenon depending on the parameters. We then connect Rényi--Sobolev inequalities to contractive and data-processing inequalities, concentration inequalities, and spectral graph theory. Our proofs in this paper are based on the information-theoretic characterization of the Rényi--Sobolev inequalities, as well as the method of types.
Paper Structure (20 sections, 10 theorems, 127 equations, 1 figure)

This paper contains 20 sections, 10 theorems, 127 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Log-Sobolev Inequalities and Log-Sobolev Functions
  3. Rényi--Sobolev Functions
  4. Our Contributions
  5. Organization
  6. Main Result
  7. Rényi--Sobolev Inequalities
  8. Binary Example
  9. Connections to Contractive and Data-Processing Inequalities
  10. Connections to Concentration Inequalities
  11. Generalized Rényi-Sobolev Function
  12. Concentration Inequalities
  13. Connections to Spectral Graph Theory
  14. Rayleigh $q$-quotient and Numerical $q$-radius
  15. Spectral Graph Theory
  16. ...and 5 more sections

Key Result

Theorem 1

For $q\in\mathbb{R}$, it holds that where ${\rm conv}\,\Xi_{q}$ denotes the lower convex envelope of the function $\Xi_{q}$. Moreover, this lower bound is asymptotically tight as $n\to\infty$, which means that If additionally, $\Xi_{q}$ is convex, then the lower bound is tight for any finite $n\ge1$. In fact, there are two distributions $(Q,R)$ and a number $\lambda\in[0,1]$ such that the sequen

Figures (1)

  • Figure 1: Illustration of the $q$-log-Sobolev function $\Xi_{q}$ for $q=0.8,1$, and $2$ (solid curves from top to bottom) and $\Xi_{q}$ for $q=0$ (dashed curve).

Theorems & Definitions (29)

  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem 1: Quasi-Tensorization polyanskiy2019improved
  • Remark 1
  • Remark 2
  • Definition 4
  • Theorem 2: Rényi--Sobolev Inequalities
  • Remark 3
  • Remark 4
  • ...and 19 more