Kernel Stein discrepancies (KSDs) measure the quality of a distributional approximation and can be computed even when the target density has an intractable normalizing constant. Notable applications include the diagnosis of approximate MCMC samplers and goodness-of-fit tests for unnormalized statistical models. The present work analyzes the convergence control properties of KSDs. We first show that standard KSDs used for weak convergence control fail to control moment convergence. To address this limitation, we next provide sufficient conditions under which alternative diffusion KSDs control both moment and weak convergence. As an immediate consequence we develop, for each $q > 0$, the first KSDs known to exactly characterize $q$-Wasserstein convergence.

翻译：核斯坦因差异（KSDs）用于衡量分布逼近的质量，即使在目标密度具有难以处理的归一化常数时仍可计算。其重要应用包括近似MCMC采样器的诊断和未归一化统计模型的拟合优度检验。本研究分析了KSDs的收敛控制特性。我们首先证明用于弱收敛控制的标准KSDs无法控制矩收敛。为克服此局限，我们进一步给出充分条件，证明在该条件下替代的扩散KSDs能够同时控制矩收敛与弱收敛。作为直接推论，我们针对每个$q > 0$构建了首个已知能精确刻画$q$-Wasserstein收敛的KSDs。