标题： 黎曼流形上的核 Stein 不相容理论与应用 显示英文标题

标题： Theory and Applications of Kernel Stein Discrepancy on Riemannian Manifolds

摘要： 分布比较是一个统计数据分析中的基础问题，在科学和工程等多个领域有着广泛的应用。 已存在许多用于分布比较的方法，但近年来核 Stein 不相容性（Kernel Stein Discrepancy, KSD）获得了极大的关注。 本文提出了一种针对黎曼流形的 KSD 的新颖且数学上严谨且一致的推广。 我们首先将 Stein 算子推广到黎曼流形，并用它在黎曼流形上建立 Stein 引理。 然后我们定义了一个新的 Stein 类，并利用它开发了所谓的合格核函数，这些核函数被用来以封闭形式定义黎曼流形上的 KSD。 我们给出了该理论在实际应用中常见的黎曼流形上的多个例子，例如齐次空间，包括 n 维球面、Grassmann 流形、Stiefel 流形、正定对称矩阵流形等。 在上述流形上，我们考虑了具有难以处理的归一化常数的各种分布，并推导出 KSD 和最小 KSD 估计量（mKSDE）的封闭表达式。 建立了 mKSDE 的若干理论性质，并且我们展示了 mKSDE 与最大似然估计（MLE）在流行的球面上的比较结果。 显示英文摘要

摘要： Distributional comparison is a fundamental problem in statistical data analysis with numerous applications in a variety of fields including but not limited to Sciences and Engineering. Numerous methods exist for distributional comparison but Kernel Stein Discrepancy (KSD) has gained significant popularity in recent times. In this paper, we present a novel mathematically rigorous and consistent generalization of KSD to Riemannian manifolds. We first generalize the Stein's operator to Riemannian manifolds and use it to establish the Stein's Lemma on Riemannian manifolds. Then we define a novel Stein class and use it to develop what we call qualified kernels that are used to define the KSD in closed form on Rimeannian manifolds. We present several examples of our theory applied to commonly encountered Riemannian manifolds in applications namely, the Riemannian homogeneous spaces, for example, the n-sphere, the Grassmanian, Stiefel, the manifold of symmetric positive definite matrices and others. On these aforementioned manifolds, we consider a variety of distributions with intractable normalization constants and derive closed form expressions for the KSD and the minimum KSD estimator (mKSDE). Several theoretical properties of mKSDE are established and we present results of comparison between mKSDE and MLE on a widely popular example, the sphere.