Dimension-free log-Sobolev inequalities for mixture distributions

TL;DR

This paper establishes that mixtures of probability measures satisfying log-Sobolev inequalities also satisfy such inequalities with dimension-free constants, including Gaussian convolutions of bounded support measures, confirming a conjecture.

Contribution

It proves that mixtures of measures with bounded chi-squared divergence inherit log-Sobolev inequalities with dimension-free constants, extending the scope of such inequalities.

Findings

Mixtures satisfy log-Sobolev inequalities with dimension-free constants.

Gaussian convolutions of bounded support measures have dimension-free log-Sobolev inequalities.

The results confirm a conjecture by Zimmermann and Bardet et al.

Abstract

We prove that if ${(P_x)}_{x\in \mathscr X}$ 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.