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

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

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Publication:2417778

DOI10.1007/978-3-319-97798-0_15zbMATH Open1414.62024arXiv1908.09211OpenAlexW2899250169MaRDI QIDQ2417778

Author name not available (Why is that?)

Publication date: 29 May 2019

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.


Full work available at URL: https://arxiv.org/abs/1908.09211



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