Group-invariant probability distributions appear in many data-generative models in machine learning, such as graphs, point clouds, and images. In practice, one often needs to estimate divergences between such distributions. In this work, we study how the inherent invariances, with respect to any smooth action of a Lie group on a manifold, improve sample complexity when estimating the Wasserstein distance, the Sobolev Integral Probability Metrics (Sobolev IPMs), the Maximum Mean Discrepancy (MMD), and also the complexity of the density estimation problem (in the $L^2$ and $L^\infty$ distance). Our results indicate a two-fold gain: (1) reducing the sample complexity by a multiplicative factor corresponding to the group size (for finite groups) or the normalized volume of the quotient space (for groups of positive dimension); (2) improving the exponent in the convergence rate (for groups of positive dimension). These results are completely new for groups of positive dimension and extend recent bounds for finite group actions.

通过研究李群对流形的平滑作用所带来的固有不变性，我们发现在估计Wasserstein距离、Sobolev积分概率度量（Sobolev IPM）、最大均值差异（MMD）以及密度估计问题的复杂性（在L^2和L^∞距离上）时，对于许多机器学习模型中出现的群不变概率分布可以大大提高样本复杂度，并改善收敛速度指数。这些结果对于正维度群是全新的，并扩展了有限群作用的最近界。

估计不变性条件下的概率差异的样本复杂性界限