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arXiv:1610.07969 (cs)
[Submitted on 25 Oct 2016]

Title:Wasserstein Stability of the Entropy Power Inequality for Log-Concave Densities

Authors:Thomas A. Courtade, Max Fathi, Ashwin Pananjady
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Abstract:We establish quantitative stability results for the entropy power inequality (EPI). Specifically, we show that if uniformly log-concave densities nearly saturate the EPI, then they must be close to Gaussian densities in the quadratic Wasserstein distance. Further, if one of the densities is log-concave and the other is Gaussian, then the deficit in the EPI can be controlled in terms of the $L^1$-Wasserstein distance. As a counterpoint, an example shows that the EPI can be unstable with respect to the quadratic Wasserstein distance when densities are uniformly log-concave on sets of measure arbitrarily close to one. Our stability results can be extended to non-log-concave densities, provided certain regularity conditions are met. The proofs are based on optimal transportation.
Comments: 19 pages
Subjects: Information Theory (cs.IT); Functional Analysis (math.FA); Probability (math.PR)
Cite as: arXiv:1610.07969 [cs.IT]
  (or arXiv:1610.07969v1 [cs.IT] for this version)
  https://doi.org/10.48550/arXiv.1610.07969

Submission history

From: Thomas Courtade [view email]
[v1] Tue, 25 Oct 2016 17:15:48 UTC (21 KB)
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