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Weak Functional Inequalities for Perturbed Measures

Patrick Cattiaux, Paula Cordero-Encinar, Arnaud Guillin

Abstract

This paper is a follow up to an article by two of the authors dedicated to the study of Poincaré and logarithmic Sobolev inequalities for measures of the form $dμ= e^{-U} dν$ where $e^{-U}$ is seen as a perturbation of $dν$. Application to the same functional inequalities for convolution products are then discussed. In the present paper we investigate similar problems for weaker functional inequalities, namely weak Poincaré, weighted Poincaré, weak log-Sobolev and weighted log-Sobolev inequalities.

Weak Functional Inequalities for Perturbed Measures

Abstract

This paper is a follow up to an article by two of the authors dedicated to the study of Poincaré and logarithmic Sobolev inequalities for measures of the form where is seen as a perturbation of . Application to the same functional inequalities for convolution products are then discussed. In the present paper we investigate similar problems for weaker functional inequalities, namely weak Poincaré, weighted Poincaré, weak log-Sobolev and weighted log-Sobolev inequalities.
Paper Structure (13 sections, 23 theorems, 114 equations)

This paper contains 13 sections, 23 theorems, 114 equations.

Table of Contents

  1. Introduction
  2. Weak Poincaré inequalities and sub-exponential relaxation
  3. Weighted Poincaré inequalities and weighted generators
  4. Why weak/weighted inequalities matter for diffusion-based generative modeling
  5. Outline and contributions
  6. Perturbation of Weak Poincaré inequalities.
  7. Weighted Poincaré inequalities.
  8. More on the links between weak and weighted Poincaré inequalities.
  9. Weaker Logarithmic Sobolev inequalities.
  10. Proofs of Section 2
  11. Proofs of Section 3
  12. Proofs of Section 4
  13. Proofs of Section 5

Key Result

Proposition 2.1

It holds

Theorems & Definitions (48)

  • Proposition 2.1
  • Proposition 2.2
  • Example 2.3
  • Proposition 2.4
  • Theorem 2.5
  • proof
  • Remark 2.6
  • Example 2.7
  • Theorem 2.8
  • Remark 2.9
  • ...and 38 more