Squared chaotic random variables: new moment inequalities with applications

On Bernstein Type Exponential Inequalities for Matrix Martingales

In this work, Bernstein's concentration inequalities for squared integrable matrix-valued discrete-time martingales are obtained. Based on Lieb's theory and Bernstein's condition, a suitable supermartingale can be constructed. Our proof is largely based on this new exponential supermartingale, Freedman's method, and Doob's stopping theorem. Our result can be regarded as an extension of Tropp's work (ECP, 2012).

Dimension-free log-Sobolev inequalities for mixture distributions

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 $μ$ is any mixing distribution on $\mathscr X$, then the mixture $\int P_x \, \mathrm{d} μ(x)$ 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.

紧群中的熵功率不等式

Entropy power inequalities in compact groups

该研究针对紧阿贝尔群上的独立随机变量，刻画两类熵功率型不等式的等号情形与稳定性估计，推导随机游走收敛到均匀分布的指数速率。

通过维数对数 Sobolev 不等式对概率测度中的低维结构进行精确检测

Sharp detection of low-dimensional structure in probability measures via dimensional logarithmic Sobolev inequalities

针对高维概率测度低维结构识别问题，提出基于维数对数 Sobolev 不等式的方法，扩展 KL 散度上界最小化工作，揭示其与假设近似的联系，在高斯与非高斯场景下均实现更优降维效果。