Relation between the Kantorovich-Wasserstein metric and the Kullback-Leibler divergence

Belavkin, Roman V. (2018) Relation between the Kantorovich-Wasserstein metric and the Kullback-Leibler divergence. In: IGAIA IV 2016: Information Geometry and Its Applications, 12-17 June 2016, Liblice, Czech Republic. (doi:https://doi.org/10.1007/978-3-319-97798-0_15)

[img]
Preview
PDF - Final accepted version (with author's formatting)
Download (208kB) | Preview

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.

Item Type: Conference or Workshop Item (Paper)
Additional Information: Paper published as: Belavkin R.V. (2018) Relation Between the Kantorovich–Wasserstein Metric and the Kullback–Leibler Divergence. In: Ay N., Gibilisco P., Matúš F. (eds) Information Geometry and Its Applications. IGAIA IV 2016. Springer Proceedings in Mathematics & Statistics, vol 252. Springer, Cham
Research Areas: A. > School of Science and Technology > Computer Science > Artificial Intelligence group
Item ID: 25970
Notes on copyright: The final authenticated version is available online at https://doi.org/10.1007/978-3-319-97798-0_15
Useful Links:
Depositing User: Roman Belavkin
Date Deposited: 11 Jan 2019 16:32
Last Modified: 30 Aug 2019 03:27
URI: https://eprints.mdx.ac.uk/id/eprint/25970

Actions (login required)

Edit Item Edit Item

Full text downloads (NB count will be zero if no full text documents are attached to the record)

Downloads per month over the past year