Dimension-free log-Sobolev inequalities for mixture distributions

We prove that if ${(P_x)}_{xin mathscr X}$ 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 $mathscr X$, 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.