标题： 黎曼流形上核斯坦方法的理论与应用 显示英文标题

标题： Theory and Applications of Kernel Stein's Method on Riemannian Manifolds

摘要： 分布比较是统计数据分析中的一个基础问题，在众多科学和工程领域有着广泛的应用。 已有多种方法用于分布比较，但近年来核 Stein 方法获得了极大的关注。 本文首先提出了一种针对黎曼流形的 Stein 算子的新颖且严格的数学一致推广。 然后我们证明，通过该算子定义的核 Stein 不相容性（KSD）在区分目标分布与参考分布方面几乎与欧几里得设置下的 KSD 一样强大。 我们研究了最小核 Stein 不相容性估计量（MKSDE）的渐近性质，将其应用于拟合优度检验，并通过实验将其与最大似然估计量（MLE）进行比较。 我们展示了该理论在实际中常见的几种黎曼流形上的应用实例，包括 n-球面、Grassmann 流形、Stiefel 流形、对称正定矩阵流形以及其他黎曼齐次空间。 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，并推导出了 KSD 和 MKSDE 的闭式表达式。 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 scientific and engineering fields. Numerous methods exist for distributional comparison but kernel Stein's method has gained significant popularity in recent times. In this paper, we first present a novel mathematically rigorous and consistent generalization of the Stein operator to Riemannian manifolds. Then we show that the kernel Stein discrepancy (KSD) defined via this operator is nearly as strong as the KSD in the Euclidean setting in terms of distinguishing the target distributions from the reference. We investigate the asymptotic properties of the minimum kernel Stein discrepancy estimator (MKSDE), apply it to goodness-of-fit testing, and compare it to the maximum likelihood estimator (MLE) experimentally. We present several examples of our theory applied to commonly encountered Riemannian manifolds in practice namely, the n-sphere, the Grassmann, Stiefel, the manifold of symmetric positive definite matrices and other Riemannian homogeneous spaces. On the aforementioned manifolds, we consider a variety of distributions with intractable normalization constants and derive closed form expressions for the KSD and MKSDE.