发表机构
Karlsruhe Institute of Technology; London School of Economics（卡尔斯鲁厄理工学院; 伦敦政治经济学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究核差异估计（MMD、HSIC、KSD）在一般拓扑空间上的极小极大下界，在对核的温和假设下证明其为\(n^{-1/2}\)，解决了这些核差异最优估计问题，均值嵌入等估计也有相同速率。
AI 中文摘要
在过去20年里，核差异被用作量化分布差异的强大工具，在双样本、拟合优度和独立性检验等方面有诸多成功应用。其最快估计器在温和条件下以参数速率\(n^{-1/2}\)收敛。虽已知在有界核的严格假设下，该速率在\(\mathbb{R}^d\)上是极小极大最优的，但对于无界核在有限维欧几里得空间之外的最优性知之甚少。本文证明了在一般拓扑空间上，在对核的温和假设下，最流行的核差异（最大均值差异、希尔伯特 - 施密特独立性准则和核斯坦差异；MMD、HSIC、KSD）估计的极小极大下界是\(n^{-1/2}\)，并表明均值嵌入和中心交叉协方差算子估计也有相同速率。结果解决了这些核差异最优估计的问题。
英文摘要
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.
