Entropy factorization via curvature
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866911968294600704 |
|---|---|
| author | Caputo, Pietro Salez, Justin |
| author_facet | Caputo, Pietro Salez, Justin |
| contents | We develop a new framework for establishing approximate factorization of entropy on arbitrary probability spaces, using a geometric notion known as non-negative sectional curvature. The resulting estimates are equivalent to entropy subadditivity and generalized Brascamp-Lieb inequalities, and provide a sharp modified log-Sobolev inequality for the Gibbs sampler of several particle systems in both continuous and discrete settings. The method allows us to obtain simple proofs of known results, as well as some new inequalities. We illustrate this through various applications, including discrete Gaussian free fields on arbitrary networks, the down-up walk on uniform $n$-sets, the uniform measure over permutations, and the uniform measure on the unit sphere in $\R^n$. Our method also yields a simple, coupling-based proof of the celebrated logarithmic Sobolev inequality for Langevin diffusions in a convex potential, which is one of the most emblematic applications of the Bakry-Émery criterion. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_13457 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Entropy factorization via curvature Caputo, Pietro Salez, Justin Probability Combinatorics Functional Analysis 39B62, 60J10 We develop a new framework for establishing approximate factorization of entropy on arbitrary probability spaces, using a geometric notion known as non-negative sectional curvature. The resulting estimates are equivalent to entropy subadditivity and generalized Brascamp-Lieb inequalities, and provide a sharp modified log-Sobolev inequality for the Gibbs sampler of several particle systems in both continuous and discrete settings. The method allows us to obtain simple proofs of known results, as well as some new inequalities. We illustrate this through various applications, including discrete Gaussian free fields on arbitrary networks, the down-up walk on uniform $n$-sets, the uniform measure over permutations, and the uniform measure on the unit sphere in $\R^n$. Our method also yields a simple, coupling-based proof of the celebrated logarithmic Sobolev inequality for Langevin diffusions in a convex potential, which is one of the most emblematic applications of the Bakry-Émery criterion. |
| title | Entropy factorization via curvature |
| topic | Probability Combinatorics Functional Analysis 39B62, 60J10 |
| url | https://arxiv.org/abs/2407.13457 |