Permutation invariance is among the most common symmetry that can be
exploited to simplify complex problems in machine learning (ML). There has been
a tremendous surge of research activities in building permutation invariant ML
architectures. However, less attention is given to how to statistically test
for permutation invariance of variables in a multivariate probability
distribution where the dimension is allowed to grow with the sample size. Also,
in terms of a statistical theory, little is known about how permutation
invariance helps with estimation in reducing dimensions. In this paper, we take
a step back and examine these questions in several fundamental problems: (i)
testing the assumption of permutation invariance of multivariate distributions;
(ii) estimating permutation invariant densities; (iii) analyzing the metric
entropy of smooth permutation invariant function classes and compare them with
their counterparts without imposing permutation invariance; (iv) kernel ridge
regression of permutation invariant functions in reproducing kernel Hilbert
space. In particular, our methods for (i) and (iv) are based on a sorting trick
and (ii) is based on an averaging trick. These tricks substantially simplify
the exploitation of permutation invariance.

本文主要研究排列不变性在机器学习中的应用，特别关注在多变量概率分布中如何统计性地检测排列不变性，以及排列不变性在简化维度估计中的作用，并针对此提出了测试方法和回归方法。