Dimension-free log-Sobolev inequalities for mixture distributions

Abstract We prove that if ( P x ) 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 ∫ P x d μ ( 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 dimension-free log-Sobolev inequalities.