Dimension-free log-Sobolev inequalities for mixture distributions

Abstract

We prove that if ${(P_x)}_{x\in \mathcal{X}}$ is a family of probability measures which satisfy the log-Sobolev inequality and whose pairwise chi-squared divergences are uniformly bounded, and $\mu$ is any mixing distribution on $\mathcal{X}$, then the mixture $\int P_x\mathrm{d}\mu (x)$ 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.

16 pages