We rigorously quantify the improvement in the sample complexity of variational divergence estimations for group-invariant distributions. In the cases of the Wasserstein-1 metric and the Lipschitz-regularized $\alpha$-divergences, the reduction of sample complexity is proportional to an ambient-dimension-dependent power of the group size. For the maximum mean discrepancy (MMD), the improvement of sample complexity is more nuanced, as it depends on not only the group size but also the choice of kernel. Numerical simulations verify our theories.

翻译：我们严格量化了变分散度估计在群不变分布下样本复杂度的改进。对于Wasserstein-1度量与Lipschitz正则化的$\alpha$-散度的情况，样本复杂度的降低与群大小的环境维度相关幂次成正比。对于最大均值差异（MMD），样本复杂度的改进更为微妙，因其不仅取决于群大小，还依赖于核函数的选择。数值模拟验证了我们的理论。