Rényi--Sobolev Inequalities and Connections to Spectral Graph Theory
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866914887549059072 |
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| author | Yu, Lei Wu, Hao |
| author_facet | Yu, Lei Wu, Hao |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_12288 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Rényi--Sobolev Inequalities and Connections to Spectral Graph Theory Yu, Lei Wu, Hao Probability Information Theory Combinatorics Functional Analysis 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. |
| title | Rényi--Sobolev Inequalities and Connections to Spectral Graph Theory |
| topic | Probability Information Theory Combinatorics Functional Analysis |
| url | https://arxiv.org/abs/2306.12288 |