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Quantitative Stability of the Entropy Power Inequality | IEEE Journals & Magazine | IEEE Xplore
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Quantitative Stability of the Entropy Power Inequality


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,...Show More

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 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 L1-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.
Published in: IEEE Transactions on Information Theory ( Volume: 64, Issue: 8, August 2018)
Page(s): 5691 - 5703
Date of Publication: 20 February 2018

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