TY - JOUR
T1 - Dimension-free log-Sobolev inequalities for mixture distributions
AU - Chen, Hong Bin
AU - Chewi, Sinho
AU - Niles-Weed, Jonathan
N1 - Funding Information:
Sinho Chewi was supported by the Department of Defense (DoD) through the National Defense Science & Engineering Graduate Fellowship (NDSEG) Program. Jonathan Niles-Weed was supported in part by National Science Foundation (NSF) grant DMS-2015291 .
Publisher Copyright:
© 2021 Elsevier Inc.
PY - 2021/12/1
Y1 - 2021/12/1
N2 - 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 ∫Pxdμ(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.
AB - 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 ∫Pxdμ(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.
KW - Dimension-free
KW - Log-Sobolev inequality
KW - Mixture distribution
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U2 - 10.1016/j.jfa.2021.109236
DO - 10.1016/j.jfa.2021.109236
M3 - Article
AN - SCOPUS:85115181965
VL - 281
JO - Journal of Functional Analysis
JF - Journal of Functional Analysis
SN - 0022-1236
IS - 11
M1 - 109236
ER -