证明了三种核分歧估计的最优收敛速度为n^{-1/2}。
Minimax Lower Bounds of Kernel Discrepancy Estimation: MMD, HSIC, KSD
- 在一般拓扑空间下，基于核函数的分歧估计达到理论最优速率。
- 该速率适用于MMD、HSIC、KSD等主流核分歧指标。
- 结果对分布嵌入与协方差算子估计同样成立，适合统计学习研究者。
过去20年，核分歧被广泛用于量化分布间的差异，在两样本检验、拟合优度检验和独立性检验中取得诸多成功。其最快估计器在温和条件下以参数率n^{-1/2}收敛。尽管在有限维欧氏空间且核有界时该速率已被证明为极小极大最优，但在更一般的设定（如无界核）下其最优性仍不明确。本文证明，在一般拓扑空间上，最常用的核分歧（最大均值差异、希尔伯特-施密特独立性判据、核斯坦分歧；MMD、HSIC、KSD）的极小极大下界为n^{-1/2}，且在核的温和假设下成立。作为推论，该速率亦适用于均值嵌入与中心化交叉协方差算子的估计。本工作解决了这些核分歧最优估计的理论问题。
原文摘要 · Abstract (English)
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.
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