Quantitative Stability of the Entropy Power Inequality

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 Kantorovich-Wasserstein distance. Furthermore, if one of the densi...

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Bibliographic Details
Published inIEEE transactions on information theory Vol. 64; no. 8; pp. 5691 - 5703
Main Authors Courtade, Thomas A., Fathi, Max, Pananjady, Ashwin
Format Journal Article
LanguageEnglish
Published New York IEEE 01.08.2018
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subjects
Atmospheric measurements
Covariance matrices
Density measurement
Entropy
Entropy power inequality
Information theory
optimal transport
Particle measurements
Stability
Transportation
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Summary: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 Kantorovich-Wasserstein distance. Furthermore, if one of the densities is Gaussian and the other is log-concave, or more generally has positive spectral gap, then the deficit in the EPI can be controlled in terms of the <inline-formula> <tex-math notation="LaTeX">L^{1} </tex-math></inline-formula>-Kantorovich-Wasserstein distance or relative entropy, respectively. As a counterpoint, an example shows that the EPI can be unstable with respect to the quadratic Kantorovich-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 mass transportation.
ISSN:0018-9448
1557-9654
DOI:10.1109/TIT.2018.2808161