Entropic curvature not comparable to other curvatures -- or is it?
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866913764746461184 |
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| author | Kamtue, Supanat Liu, Shiping Münch, Florentin Peyerimhoff, Norbert |
| author_facet | Kamtue, Supanat Liu, Shiping Münch, Florentin Peyerimhoff, Norbert |
| contents | In this paper we consider global $θ$-curvatures of finite Markov chains with associated means $θ$ in the spirit of the entropic curvature (based on the logarithmic mean) by Erbar-Maas and Mielke. As in the case of Bakry-Émery curvature, we also allow for a finite dimension parameter by making use of an adapted $Γ$ calculus for $θ$-curvatures. We prove explicit positive lower curvature bounds (both finite- and infinite-dimensional) for finite abelian Cayley graphs. In the case of cycles, we provide also an upper curvature bound which shows that our lower bounds are asymptotically sharp (up to a logarithmic factor). Moreover, we prove new universal lower curvature bounds for finite Markov chains as well as curvature perturbation results (allowing, in particular, to compare entropic and Bakry-Émery curvatures). Finally, we present examples where entropic curvature differs significantly from other curvature notions like Bakry-Émery curvature or Ollivier Ricci and sectional curvatures. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_04581 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Entropic curvature not comparable to other curvatures -- or is it? Kamtue, Supanat Liu, Shiping Münch, Florentin Peyerimhoff, Norbert Differential Geometry Metric Geometry Probability 53C21, 60J10, 05C81 In this paper we consider global $θ$-curvatures of finite Markov chains with associated means $θ$ in the spirit of the entropic curvature (based on the logarithmic mean) by Erbar-Maas and Mielke. As in the case of Bakry-Émery curvature, we also allow for a finite dimension parameter by making use of an adapted $Γ$ calculus for $θ$-curvatures. We prove explicit positive lower curvature bounds (both finite- and infinite-dimensional) for finite abelian Cayley graphs. In the case of cycles, we provide also an upper curvature bound which shows that our lower bounds are asymptotically sharp (up to a logarithmic factor). Moreover, we prove new universal lower curvature bounds for finite Markov chains as well as curvature perturbation results (allowing, in particular, to compare entropic and Bakry-Émery curvatures). Finally, we present examples where entropic curvature differs significantly from other curvature notions like Bakry-Émery curvature or Ollivier Ricci and sectional curvatures. |
| title | Entropic curvature not comparable to other curvatures -- or is it? |
| topic | Differential Geometry Metric Geometry Probability 53C21, 60J10, 05C81 |
| url | https://arxiv.org/abs/2404.04581 |