标题： 核Stein差异估计的极小极大下界 显示英文标题

标题： The Minimax Lower Bound of Kernel Stein Discrepancy Estimation

摘要： 核Stein偏差（KSDs）在过去十年中已成为一种强大的工具，用于量化拟合优度，具有许多成功的应用。据我们所知，所有现有已知率的KSD估计器都达到$\sqrt n$收敛。在本工作中，我们提出了两个互补的结果（采用不同的证明策略），确立了KSD估计的最小最大下界为$n^{-1/2}$，并确定了这些估计器的最优性。我们的第一个结果关注使用Langevin-Stein算子的$\mathbb R^d$上的KSD估计；我们对高斯核的显式常数表明，KSD估计的难度可能随着维度$d$呈指数增长。我们的第二个结果确定了在一般域上KSD估计的最小最大下界。 显示英文摘要

摘要： Kernel Stein discrepancies (KSDs) have emerged as a powerful tool for quantifying goodness-of-fit over the last decade, featuring numerous successful applications. To the best of our knowledge, all existing KSD estimators with known rate achieve $\sqrt n$-convergence. In this work, we present two complementary results (with different proof strategies), establishing that the minimax lower bound of KSD estimation is $n^{-1/2}$ and settling the optimality of these estimators. Our first result focuses on KSD estimation on $\mathbb R^d$ with the Langevin-Stein operator; our explicit constant for the Gaussian kernel indicates that the difficulty of KSD estimation may increase exponentially with the dimensionality $d$. Our second result settles the minimax lower bound for KSD estimation on general domains.