Understanding two-scale criteria for Poincaré and log-Sobolev inequalities in the Euclidean case through Φ-entropies
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915502547271680 |
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| author | Srinivasan, Vishwak |
| author_facet | Srinivasan, Vishwak |
| contents | We study settings in which mixture and joint distributions satisfy a Poincaré (or log-Sobolev) inequality induced by a marginal and a collection of conditional distributions that are assumed to satisfy Poincaré (or log-Sobolev, resp.) inequalities and supported over Euclidean spaces. In this note, we use the framework of $Φ$-Sobolev inequalities (Chafaï, 2004) to provide a unified approach to arriving at these inequalities in the Euclidean case. This results in a simpler proof technique for establishing these functional inequalities under certain two-scale criteria. We also discuss applications of these results to certain sampling algorithms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_15410 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Understanding two-scale criteria for Poincaré and log-Sobolev inequalities in the Euclidean case through Φ-entropies Srinivasan, Vishwak Probability Functional Analysis We study settings in which mixture and joint distributions satisfy a Poincaré (or log-Sobolev) inequality induced by a marginal and a collection of conditional distributions that are assumed to satisfy Poincaré (or log-Sobolev, resp.) inequalities and supported over Euclidean spaces. In this note, we use the framework of $Φ$-Sobolev inequalities (Chafaï, 2004) to provide a unified approach to arriving at these inequalities in the Euclidean case. This results in a simpler proof technique for establishing these functional inequalities under certain two-scale criteria. We also discuss applications of these results to certain sampling algorithms. |
| title | Understanding two-scale criteria for Poincaré and log-Sobolev inequalities in the Euclidean case through Φ-entropies |
| topic | Probability Functional Analysis |
| url | https://arxiv.org/abs/2509.15410 |