🤖 AI Summary
This study establishes minimax lower bounds for the estimation of Maximum Mean Discrepancy (MMD), Hilbert–Schmidt Independence Criterion (HSIC), and Kernelized Stein Discrepancy (KSD) in general topological spaces under unbounded kernel conditions. By integrating reproducing kernel Hilbert space theory, functional analysis, and a minimax information-theoretic framework, the work rigorously proves—under mild assumptions—that the optimal convergence rate for these three classes of kernel-based discrepancy measures remains $n^{-1/2}$. This result resolves a long-standing open theoretical question and extends to the estimation of mean embeddings and centered cross-covariance operators, thereby establishing the minimax optimality of their parametric convergence rates.
📝 Abstract
Over the past 20 years, kernel discrepancies have been leveraged as a highly powerful tool for quantifying the disagreement of distributions, with numerous successful applications in two-sample, goodness-of-fit, and independence testing, among others. Their fastest estimators are known to converge at a parametric rate---$n^{-1/2}$---under mild conditions. While this rate is known to be minimax optimal on $\mathbb R^d$ under strict assumptions with bounded kernels, little is known about its optimality beyond the finite-dimensional Euclidean setting with unbounded kernels. In this work, we prove that the minimax lower bound of estimation of the most popular kernel discrepancies (maximum mean discrepancy, Hilbert-Schmidt independence criterion and kernel Stein discrepancy; MMD, HSIC, KSD) is $n^{-1/2}$ on general topological spaces, and under mild assumptions on the kernel; the same rates are shown (as corollaries) to hold for the estimation of the mean embedding and the centered cross-covariance operator. Our results settle the question of optimal estimation of these kernel discrepancies.