We explore the minimax optimality of goodness-of-fit tests on general domains using the kernelized Stein discrepancy (KSD). The KSD framework offers a flexible approach for goodness-of-fit testing, avoiding strong distributional assumptions, accommodating diverse data structures beyond Euclidean spaces, and relying only on partial knowledge of the reference distribution, while maintaining computational efficiency. We establish a general framework and an operator-theoretic representation of the KSD, encompassing many existing KSD tests in the literature, which vary depending on the domain. We reveal the characteristics and limitations of KSD and demonstrate its non-optimality under a certain alternative space, defined over general domains when considering $\chi^2$-divergence as the separation metric. To address this issue of non-optimality, we propose a modified, minimax optimal test by incorporating a spectral regularizer, thereby overcoming the shortcomings of standard KSD tests. Our results are established under a weak moment condition on the Stein kernel, which relaxes the bounded kernel assumption required by prior work in the analysis of kernel-based hypothesis testing. Additionally, we introduce an adaptive test capable of achieving minimax optimality up to a logarithmic factor by adapting to unknown parameters. Through numerical experiments, we illustrate the superior performance of our proposed tests across various domains compared to their unregularized counterparts.

翻译：我们探索了使用核斯坦因散度（KSD）在一般领域上进行拟合优度检验的最小最大最优性。KSD框架为拟合优度检验提供了一种灵活的方法，避免了严格的分布假设，适应了欧几里得空间之外多样化的数据结构，仅依赖于参考分布的部分知识，同时保持了计算效率。我们建立了一个通用框架和KSD的算子理论表示，涵盖了文献中许多现有的KSD检验，这些检验因领域不同而有所变化。我们揭示了KSD的特性和局限性，并证明了在考虑χ²散度作为分离度量时，在一般领域上定义的特定备择空间下，KSD的非最优性。为了解决这一非最优性问题，我们通过引入光谱正则化器提出了一种改进的最小最大最优检验，从而克服了标准KSD检验的缺陷。我们的结果是在斯坦因核的弱矩条件下建立的，这放宽了先前基于核的假设检验分析中所需的有界核假设。此外，我们引入了一种自适应检验，能够通过适应未知参数，在对数因子范围内实现最小最大最优性。通过数值实验，我们展示了我们提出的检验在各种领域上相比未正则化的对应检验具有更优越的性能。