Dimension-free log-Sobolev inequalities for mixture distributions

By Hong-Bin Chen, Sinho Chewi, Jonathan Niles-Weed. Journal of Functional Analysis, 2021

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We prove that if (Px)x∈X is a family of probability measures which satisfy the log-Sobolev inequality and whose pairwise chi-squared divergences are uniformly bounded, and μ is any mixing distribution on X , then the mixture ∫ Px dμ(x) satisfies a log-Sobolev inequality. In various settings of interest, the resulting log-Sobolev constant is dimension-free. In particular, our result implies a conjecture of Zimmermann and Bardet et al. that Gaussian convolutions of measures with bounded support enjoy dimensionfree log-Sobolev inequalities.

Subfields: Applied Mathematics

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