Dimension-free log-Sobolev inequalities for mixture distributions
Dimension-free log-Sobolev inequalities for mixture distributions
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DOI：
10.1016/j.jfa.2021.109236
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发表时间：
2021-02
影响因子：
1.7
通讯作者：
Hong-Bin Chen;Sinho Chewi;Jonathan Niles-Weed
中科院分区：
文献类型：
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作者：
Hong-Bin Chen;Sinho Chewi;Jonathan Niles-Weed
证明了如果（P x） x∈x是满足log-Sobolev不等式且其成对卡方散度一致有界的概率测度族，且μ是x上的任意混合分布，则混合∫P x d μ (x)满足log-Sobolev不等式。在不同的情况下，得到的对数索博列夫常数是无维的。特别地，我们的结果暗示了Zimmermann和Bardet等人的一个猜想，即具有有界支持的测度的高斯卷积具有无维log-Sobolev不等式。
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.