🤖 AI Summary
This study addresses the minimax optimal estimation of Kernel Stein Discrepancy (KSD) when only the score function of the target distribution and a finite sample are available. By analyzing the spectral properties of the Stein covariance operator, the authors establish for the first time that the minimax risk of KSD estimation is governed by the Hilbert–Schmidt norm of this operator, yielding an optimal convergence rate of √(|C⋆|_HS / n). They propose a positive-part square-root U-statistic estimator that achieves this optimal rate. In contrast, the conventional V-statistic estimator attains only √(tr(C⋆) / n), resulting in an exponentially larger error gap in high-dimensional Gaussian settings, thereby highlighting the substantial advantage of the proposed method.
📝 Abstract
Kernel Stein Discrepancy (KSD) compares a sample to a fixed target distribution known only through its score, and is widely used for goodness-of-fit testing, sample quality assessment, and approximate inference. We study the estimation of $\operatorname{KSD}(P_0,P)$ from $n$ independent observations and identify the sharp spectral constant governing the minimax risk: it is the Hilbert-Schmidt norm of the Stein covariance operator $C_\star$, giving the minimax scale $\sqrt{\|C_\star\|_{\mathrm{HS}}/n}$. This scale is attained by the positive-part square-root U-statistic, whereas the standard plug-in V-statistic remains at the trace scale $\sqrt{\operatorname{tr}(C_\star)/n}$ and is therefore suboptimal by the fourth root of the effective rank of $C_\star$; for a Gaussian target with a fixed-bandwidth Gaussian kernel this factor is exponential in the dimension.