Concentration of measure on spheres and related manifolds
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , |
|---|---|
| Format: | Preprint |
| Publié: |
2024
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866914905739755520 |
|---|---|
| author | Götze, Friedrich Sambale, Holger |
| author_facet | Götze, Friedrich Sambale, Holger |
| contents | We study various generalizations of concentration of measure on the unit sphere, in particular by means of log-Sobolev inequalities. First, we show Sudakov-type concentration results and local semicircular laws for weighted random matrices. A further branch addresses higher order concentration (i.\,e., concentration for non-Lipschitz functions which have bounded derivatives of higher order) for $\ell_p^n$-spheres. This is based on a type of generalized log-Sobolev inqualities referred to as $\mathrm{LS}_q$-inequalities. More generally, we prove higher order concentration bounds for probability measures on $\mathbb{R}^n$ which satisfy an $\mathrm{LS}_q$-inequality. Finally, we derive concentration bounds for sequences of smooth symmetric functions on the Euclidean sphere which are closely related to Edgeworth-type expansions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_04346 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Concentration of measure on spheres and related manifolds Götze, Friedrich Sambale, Holger Probability Functional Analysis 60B20, 60E15, 60F10 We study various generalizations of concentration of measure on the unit sphere, in particular by means of log-Sobolev inequalities. First, we show Sudakov-type concentration results and local semicircular laws for weighted random matrices. A further branch addresses higher order concentration (i.\,e., concentration for non-Lipschitz functions which have bounded derivatives of higher order) for $\ell_p^n$-spheres. This is based on a type of generalized log-Sobolev inqualities referred to as $\mathrm{LS}_q$-inequalities. More generally, we prove higher order concentration bounds for probability measures on $\mathbb{R}^n$ which satisfy an $\mathrm{LS}_q$-inequality. Finally, we derive concentration bounds for sequences of smooth symmetric functions on the Euclidean sphere which are closely related to Edgeworth-type expansions. |
| title | Concentration of measure on spheres and related manifolds |
| topic | Probability Functional Analysis 60B20, 60E15, 60F10 |
| url | https://arxiv.org/abs/2408.04346 |