Bakry-\'Emery calculus for entropic curvature, new diameter estimates, and spectral gaps
In this paper, we propose a generalization of Bakry-\'Emery's calculus which allows us to formulate both Bakry-\'Emery and entropic curvature simultaneously. This formulation represents both curvatures as an integral of the Bochner formula against some measure. This leads to a natural...
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| creator | Kamtue, Supanat Liu, Shiping Münch, Florentin Peyerimhoff, Norbert |
| description | In this paper, we propose a generalization of Bakry-\'Emery's calculus which
allows us to formulate both Bakry-\'Emery and entropic curvature
simultaneously. This formulation represents both curvatures as an integral of
the Bochner formula against some measure. This leads to a natural optimality
criterion of measures, which we investigate in the Bakry-\'Emery setting.
Moreover, our approach leads also to a dimension parameter in the framework of
entropic curvature.
We also present gradient estimate applications, that is, diameter estimates
for Markov chains with strictly positive entropic curvature and a spectral gap
estimate. The latter implies non-existence of non-negatively curved expanders. |
| doi_str_mv | 10.48550/arxiv.2312.09686 |
| format | Article |
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allows us to formulate both Bakry-\'Emery and entropic curvature
simultaneously. This formulation represents both curvatures as an integral of
the Bochner formula against some measure. This leads to a natural optimality
criterion of measures, which we investigate in the Bakry-\'Emery setting.
Moreover, our approach leads also to a dimension parameter in the framework of
entropic curvature.
We also present gradient estimate applications, that is, diameter estimates
for Markov chains with strictly positive entropic curvature and a spectral gap
estimate. The latter implies non-existence of non-negatively curved expanders.</description><identifier>DOI: 10.48550/arxiv.2312.09686</identifier><language>eng</language><subject>Mathematics - Combinatorics ; Mathematics - Differential Geometry</subject><creationdate>2023-12</creationdate><rights>http://arxiv.org/licenses/nonexclusive-distrib/1.0</rights><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>228,230,780,885</link.rule.ids><linktorsrc>$$Uhttps://arxiv.org/abs/2312.09686$$EView_record_in_Cornell_University$$FView_record_in_$$GCornell_University$$Hfree_for_read</linktorsrc><backlink>$$Uhttps://doi.org/10.48550/arXiv.2312.09686$$DView paper in arXiv$$Hfree_for_read</backlink></links><search><creatorcontrib>Kamtue, Supanat</creatorcontrib><creatorcontrib>Liu, Shiping</creatorcontrib><creatorcontrib>Münch, Florentin</creatorcontrib><creatorcontrib>Peyerimhoff, Norbert</creatorcontrib><title>Bakry-\'Emery calculus for entropic curvature, new diameter estimates, and spectral gaps</title><description>In this paper, we propose a generalization of Bakry-\'Emery's calculus which
allows us to formulate both Bakry-\'Emery and entropic curvature
simultaneously. This formulation represents both curvatures as an integral of
the Bochner formula against some measure. This leads to a natural optimality
criterion of measures, which we investigate in the Bakry-\'Emery setting.
Moreover, our approach leads also to a dimension parameter in the framework of
entropic curvature.
We also present gradient estimate applications, that is, diameter estimates
for Markov chains with strictly positive entropic curvature and a spectral gap
estimate. The latter implies non-existence of non-negatively curved expanders.</description><subject>Mathematics - Combinatorics</subject><subject>Mathematics - Differential Geometry</subject><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2023</creationdate><recordtype>article</recordtype><sourceid>GOX</sourceid><recordid>eNotjztPwzAYRb0woMIPYMIbSxP8tjNCVaBSJZYODEjRFz9QRJJGtlPIvycUlnuXq6N7ELqhpBRGSnIP8bs9lYxTVpJKGXWJ3h7hM87F-92293HGFjo7dVPC4RixH3I8jq3FdoonyFP0azz4L-xa6H32yyDltofs0xrD4HAavc0ROvwBY7pCFwG65K__e4UOT9vD5qXYvz7vNg_7ApRWhXSCm8CBSOp8oIQ1CsKSlRGgpKa6EVw2QVhVgQYDjDmqnSCOWlNV1vAVuv3DntXqMS6H4lz_KtZnRf4DGABMhA</recordid><startdate>20231215</startdate><enddate>20231215</enddate><creator>Kamtue, Supanat</creator><creator>Liu, Shiping</creator><creator>Münch, Florentin</creator><creator>Peyerimhoff, Norbert</creator><scope>AKZ</scope><scope>GOX</scope></search><sort><creationdate>20231215</creationdate><title>Bakry-\'Emery calculus for entropic curvature, new diameter estimates, and spectral gaps</title><author>Kamtue, Supanat ; Liu, Shiping ; Münch, Florentin ; Peyerimhoff, Norbert</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-a676-5d438f3a051def102b6af02b984a65717b435bf4c69a7a8a22d17d40d1c899c83</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2023</creationdate><topic>Mathematics - Combinatorics</topic><topic>Mathematics - Differential Geometry</topic><toplevel>online_resources</toplevel><creatorcontrib>Kamtue, Supanat</creatorcontrib><creatorcontrib>Liu, Shiping</creatorcontrib><creatorcontrib>Münch, Florentin</creatorcontrib><creatorcontrib>Peyerimhoff, Norbert</creatorcontrib><collection>arXiv Mathematics</collection><collection>arXiv.org</collection></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext_linktorsrc</fulltext></delivery><addata><au>Kamtue, Supanat</au><au>Liu, Shiping</au><au>Münch, Florentin</au><au>Peyerimhoff, Norbert</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Bakry-\'Emery calculus for entropic curvature, new diameter estimates, and spectral gaps</atitle><date>2023-12-15</date><risdate>2023</risdate><abstract>In this paper, we propose a generalization of Bakry-\'Emery's calculus which
allows us to formulate both Bakry-\'Emery and entropic curvature
simultaneously. This formulation represents both curvatures as an integral of
the Bochner formula against some measure. This leads to a natural optimality
criterion of measures, which we investigate in the Bakry-\'Emery setting.
Moreover, our approach leads also to a dimension parameter in the framework of
entropic curvature.
We also present gradient estimate applications, that is, diameter estimates
for Markov chains with strictly positive entropic curvature and a spectral gap
estimate. The latter implies non-existence of non-negatively curved expanders.</abstract><doi>10.48550/arxiv.2312.09686</doi><oa>free_for_read</oa></addata></record> |
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| subjects | Mathematics - Combinatorics Mathematics - Differential Geometry |
| title | Bakry-\'Emery calculus for entropic curvature, new diameter estimates, and spectral gaps |
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