Symplectic Bregman Divergences
We present a generalization of Bregman divergences in finite-dimensional symplectic vector spaces that we term symplectic Bregman divergences. Symplectic Bregman divergences are derived from a symplectic generalization of the Fenchel–Young inequality which relies on the notion of symplectic subdiffe...
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2024-12-01
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| author | Frank Nielsen |
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| description | We present a generalization of Bregman divergences in finite-dimensional symplectic vector spaces that we term symplectic Bregman divergences. Symplectic Bregman divergences are derived from a symplectic generalization of the Fenchel–Young inequality which relies on the notion of symplectic subdifferentials. The symplectic Fenchel–Young inequality is obtained using the symplectic Fenchel transform which is defined with respect to the symplectic form. Since symplectic forms can be built generically from pairings of dual systems, we obtain a generalization of Bregman divergences in dual systems obtained by equivalent symplectic Bregman divergences. In particular, when the symplectic form is derived from an inner product, we show that the corresponding symplectic Bregman divergences amount to ordinary Bregman divergences with respect to composite inner products. Some potential applications of symplectic divergences in geometric mechanics, information geometry, and learning dynamics in machine learning are touched upon. |
| format | Article |
| id | doaj-art-3b94c0f96fd5413aa1c53b8ae28f5f56 |
| institution | Kabale University |
| issn | 1099-4300 |
| language | English |
| publishDate | 2024-12-01 |
| publisher | MDPI AG |
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| series | Entropy |
| spelling | doaj-art-3b94c0f96fd5413aa1c53b8ae28f5f562024-12-27T14:25:12ZengMDPI AGEntropy1099-43002024-12-012612110110.3390/e26121101Symplectic Bregman DivergencesFrank Nielsen0Sony Computer Science Laboratories Inc., Tokyo 141-0022, JapanWe present a generalization of Bregman divergences in finite-dimensional symplectic vector spaces that we term symplectic Bregman divergences. Symplectic Bregman divergences are derived from a symplectic generalization of the Fenchel–Young inequality which relies on the notion of symplectic subdifferentials. The symplectic Fenchel–Young inequality is obtained using the symplectic Fenchel transform which is defined with respect to the symplectic form. Since symplectic forms can be built generically from pairings of dual systems, we obtain a generalization of Bregman divergences in dual systems obtained by equivalent symplectic Bregman divergences. In particular, when the symplectic form is derived from an inner product, we show that the corresponding symplectic Bregman divergences amount to ordinary Bregman divergences with respect to composite inner products. Some potential applications of symplectic divergences in geometric mechanics, information geometry, and learning dynamics in machine learning are touched upon.https://www.mdpi.com/1099-4300/26/12/1101dual systemduality productinner productsymplectic formsymplectic matrix groupsymplectic subdifferential |
| spellingShingle | Frank Nielsen Symplectic Bregman Divergences Entropy dual system duality product inner product symplectic form symplectic matrix group symplectic subdifferential |
| title | Symplectic Bregman Divergences |
| title_full | Symplectic Bregman Divergences |
| title_fullStr | Symplectic Bregman Divergences |
| title_full_unstemmed | Symplectic Bregman Divergences |
| title_short | Symplectic Bregman Divergences |
| title_sort | symplectic bregman divergences |
| topic | dual system duality product inner product symplectic form symplectic matrix group symplectic subdifferential |
| url | https://www.mdpi.com/1099-4300/26/12/1101 |
| work_keys_str_mv | AT franknielsen symplecticbregmandivergences |