A View on Optimal Transport from Noncommutative Geometry
We discuss the relation between the Wasserstein distance of order 1 between probability distributions on a metric space, arising in the study of Monge-Kantorovich transport problem, and the spectral distance of noncommutative geometry. Starting from a remark of Rieffel on compact manifolds, we first...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2010 |
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| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2010
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/146358 |
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| Cite this: | A View on Optimal Transport from Noncommutative Geometry / F. D'Andrea, P. Martinetti // Symmetry, Integrability and Geometry: Methods and Applications. — 2010. — Т. 6. — Бібліогр.: 44 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862714269423894528 |
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| author | D'Andrea, F. Martinetti, P. |
| author_facet | D'Andrea, F. Martinetti, P. |
| citation_txt | A View on Optimal Transport from Noncommutative Geometry / F. D'Andrea, P. Martinetti // Symmetry, Integrability and Geometry: Methods and Applications. — 2010. — Т. 6. — Бібліогр.: 44 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We discuss the relation between the Wasserstein distance of order 1 between probability distributions on a metric space, arising in the study of Monge-Kantorovich transport problem, and the spectral distance of noncommutative geometry. Starting from a remark of Rieffel on compact manifolds, we first show that on any - i.e. non-necessary compact - complete Riemannian spin manifolds, the two distances coincide. Then, on convex manifolds in the sense of Nash embedding, we provide some natural upper and lower bounds to the distance between any two probability distributions. Specializing to the Euclidean space Rⁿ, we explicitly compute the distance for a particular class of distributions generalizing Gaussian wave packet. Finally we explore the analogy between the spectral and the Wasserstein distances in the noncommutative case, focusing on the standard model and the Moyal plane. In particular we point out that in the two-sheet space of the standard model, an optimal-transport interpretation of the metric requires a cost function that does not vanish on the diagonal. The latest is similar to the cost function occurring in the relativistic heat equation.
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| first_indexed | 2025-12-07T17:49:28Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-146358 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T17:49:28Z |
| publishDate | 2010 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | D'Andrea, F. Martinetti, P. 2019-02-09T09:38:18Z 2019-02-09T09:38:18Z 2010 A View on Optimal Transport from Noncommutative Geometry / F. D'Andrea, P. Martinetti // Symmetry, Integrability and Geometry: Methods and Applications. — 2010. — Т. 6. — Бібліогр.: 44 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 58B34; 82C70 doi:10.3842/SIGMA.2010.057 https://nasplib.isofts.kiev.ua/handle/123456789/146358 We discuss the relation between the Wasserstein distance of order 1 between probability distributions on a metric space, arising in the study of Monge-Kantorovich transport problem, and the spectral distance of noncommutative geometry. Starting from a remark of Rieffel on compact manifolds, we first show that on any - i.e. non-necessary compact - complete Riemannian spin manifolds, the two distances coincide. Then, on convex manifolds in the sense of Nash embedding, we provide some natural upper and lower bounds to the distance between any two probability distributions. Specializing to the Euclidean space Rⁿ, we explicitly compute the distance for a particular class of distributions generalizing Gaussian wave packet. Finally we explore the analogy between the spectral and the Wasserstein distances in the noncommutative case, focusing on the standard model and the Moyal plane. In particular we point out that in the two-sheet space of the standard model, an optimal-transport interpretation of the metric requires a cost function that does not vanish on the diagonal. The latest is similar to the cost function occurring in the relativistic heat equation. This paper is a contribution to the Special Issue “Noncommutative Spaces and Fields”. The full collection is available at http://www.emis.de/journals/SIGMA/noncommutative.html.
 We would like to thank Hanfeng Li and anonymous referees for their valuable comments. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications A View on Optimal Transport from Noncommutative Geometry Article published earlier |
| spellingShingle | A View on Optimal Transport from Noncommutative Geometry D'Andrea, F. Martinetti, P. |
| title | A View on Optimal Transport from Noncommutative Geometry |
| title_full | A View on Optimal Transport from Noncommutative Geometry |
| title_fullStr | A View on Optimal Transport from Noncommutative Geometry |
| title_full_unstemmed | A View on Optimal Transport from Noncommutative Geometry |
| title_short | A View on Optimal Transport from Noncommutative Geometry |
| title_sort | view on optimal transport from noncommutative geometry |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/146358 |
| work_keys_str_mv | AT dandreaf aviewonoptimaltransportfromnoncommutativegeometry AT martinettip aviewonoptimaltransportfromnoncommutativegeometry AT dandreaf viewonoptimaltransportfromnoncommutativegeometry AT martinettip viewonoptimaltransportfromnoncommutativegeometry |