Relation between the Kantorovich-Wasserstein metric and the Kullback-Leibler divergence
Conference paper
Belavkin, R. 2018. Relation between the Kantorovich-Wasserstein metric and the Kullback-Leibler divergence. Ay, N., Gibilisco, P. and Matúš, F. (ed.) IGAIA IV 2016: Information Geometry and Its Applications. Liblice, Czech Republic 12 - 17 Jun 2016 Springer. pp. 363-373 https://doi.org/10.1007/978-3-319-97798-0_15
| Type | Conference paper |
|---|---|
| Title | Relation between the Kantorovich-Wasserstein metric and the Kullback-Leibler divergence |
| Authors | Belavkin, R. |
| Abstract | We discuss a relation between the Kantorovich-Wasserstein (KW) metric and the Kullback-Leibler (KL) divergence. The former is defined using the optimal transport problem (OTP) in the Kantorovich formulation. The latter is used to define entropy and mutual information, which appear in variational problems to find optimal channel (OCP) from the rate distortion and the value of information theories. We show that OTP is equivalent to OCP with one additional constraint fixing the output measure, and therefore OCP with constraints on the KL-divergence gives a lower bound on the KW-metric. The dual formulation of OTP allows us to explore the relation between the KL-divergence and the KW-metric using decomposition of the former based on the law of cosines. This way we show the link between two divergences using the variational and geometric principles. |
| Research Group | Artificial Intelligence group |
| Conference | IGAIA IV 2016: Information Geometry and Its Applications |
| Page range | 363-373 |
| Proceedings Title | Information Geometry and Its Applications: On the Occasion of Shun-ichi Amari's 80th Birthday, IGAIA IV Liblice, Czech Republic, June 2016 |
| Series | Springer Proceedings in Mathematics & Statistics |
| Editors | Ay, N., Gibilisco, P. and Matúš, F. |
| ISSN | 2194-1009 |
| Electronic | 2194-1017 |
| ISBN | |
| Hardcover | 9783319977973 |
| Paperback | 9783030074050 |
| Electronic | 9783319977980 |
| Publisher | Springer |
| Publication dates | |
| Online | 04 Nov 2018 |
| 01 Jan 2018 | |
| Publication process dates | |
| Deposited | 11 Jan 2019 |
| Accepted | 15 Apr 2016 |
| Output status | Published |
| Accepted author manuscript | File Access Level Open |
| Copyright Statement | The final authenticated version is available online at https://doi.org/10.1007/978-3-319-97798-0_15 |
| Additional information | Paper published as: |
| Digital Object Identifier (DOI) | https://doi.org/10.1007/978-3-319-97798-0_15 |
| Web address (URL) of conference proceedings | https://doi.org/10.1007/978-3-319-97798-0 |
| Language | English |
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