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We rigorously quantify the improvement in the sample complexity ofvariational divergence estimations for group-invariant distributions. In thecases of the Wasserstein-1 metric and the Lipschitz-regularized$\alpha$-divergences, the reduction of sample complexity is proportional to anambient-dimension-dependent power of the group size. For the maximum meandiscrepancy (MMD), the improvement of sample complexity is more nuanced, as itdepends on not only the group size but also the choice of kernel. Numericalsimulations verify our theories.

Sample Complexity of Probability Divergences under Group Symmetry