证明了核Stein散度估计的最优收敛速度为n^{-1/2}。
The Minimax Lower Bound of Kernel Stein Discrepancy Estimation
- 通过Langevin-Stein算子研究R^d上的KSD估计
- 揭示了高维下估计难度随维度指数级增长
- 为现有估计算法的最优性提供了理论依据
核Stein散度（KSD）在过去十年中成为衡量拟合优度的强大工具，已广泛应用于各类场景。据我们所知，所有具有已知收敛速率的KSD估计器均达到√n收敛。本文提出两个互补结果（采用不同证明方法），确立了KSD估计的极小极大下界为n^{-1/2}，从而证实了现有估计器的最优性。第一项结果针对R^d上基于Langevin-Stein算子的KSD估计，其对高斯核的显式常数表明，估计难度随维度d呈指数增长。第二项结果则在一般定义域上确立了KSD估计的极小极大下界。
原文摘要 · Abstract (English)
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.
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