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：核斯坦因散度（KSD）用于衡量分布近似的质量，即使目标密度含有难以计算的归一化常数也能进行计算。其重要应用包括近似MCMC采样器的诊断以及未归一化统计模型的拟合优度检验。本文分析了KSD的收敛控制特性。我们首先证明，用于弱收敛控制的标准KSD无法控制矩收敛。为解决这一局限，我们随后给出了替代性扩散KSD同时控制矩收敛与弱收敛的充分条件。作为直接推论，我们对每个$q > 0$首次提出了已知能精确刻画$q$-Wasserstein收敛性的KSD。