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：我们严格量化了群不变分布变分散度估计中样本复杂度的改进。在Wasserstein-1度量和Lipschitz正则化$\alpha$-散度的情况下，样本复杂度的降低与群大小乘以其依赖环境维度的幂次成正比。对于最大平均差异（MMD），样本复杂度的改进更为微妙，它不仅取决于群大小，还取决于核函数的选择。数值模拟验证了我们的理论。