Abstract: In this talk, we consider a product distribution with each marginal belonging to a class of probability distributions \mathcal{P}. We study the statistical distance between the permutation mixture P formed by randomly permuting the coordinates of the product distribution, and an i.i.d. product Q of the marginals of P. Under mild conditions on \mathcal{P}, we establish an upper bound of the chi-squared divergence between P and Q depending only on \mathcal{P} but not the dimension, quantifying the weak dependence in permutation mixtures. In particular, we show that the classical method of moments based on orthogonal polynomials suffers from the curse of dimensionality, and new basis functions are necessary for mixture distributions in high dimensions. Another key step is a saddle point analysis for elementary symmetric polynomials and matrix permanents, which could be of independent interest. Applications to de Finette-type results and empirical Bayes will be discussed.
Speaker's bio: I received my B.E. in Electronic Engineering from Tsinghua University in Jul 2015, and my M.S. and Ph.D. in Electrical Engineering from Stanford University in Aug 2021, under the supervision of Tsachy Weissman. I was a postdoctoral scholar at the Simons Institute for the Theory of Computing, University of California, Berkeley in 2021-22, and a Norbert Wiener postdoctoral associate at the Statistics and Data Science Center (SDSC) in MIT IDSS in 2022-23. I am broadly interested in the mathematics of data science, including statistics, learning theory, bandits, and information theory.