  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 1-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.
