We present a new framework for statistical inference on Riemannian manifolds that achieves high-order accuracy, addressing the challenges posed by non-Euclidean parameter spaces frequently encountered in modern data science. Our approach leverages a novel and computationally efficient procedure to reach higher-order asymptotic precision. In particular, we develop a bootstrap algorithm on Riemannian manifolds that is both computationally efficient and accurate for hypothesis testing and confidence region construction. Although locational hypothesis testing can be reformulated as a standard Euclidean problem, constructing high-order accurate confidence regions necessitates careful treatment of manifold geometry. To this end, we establish high-order asymptotics under an appropriate coordinate representation induced by a second-order retraction, thereby enabling precise expansions that incorporate curvature effects. We demonstrate the versatility of this framework across various manifold settings, including spheres, the Stiefel manifold, fixed-rank matrix manifolds, and rank-one tensor manifolds; for Euclidean submanifolds, we also introduce a class of projection-like coordinate charts with strong consistency properties. Finally, numerical studies confirm the practical merits of the proposed procedure.

翻译：本文提出了一种在黎曼流形上进行统计推断的新框架，该框架实现了高阶精确度，解决了现代数据科学中经常遇到的非欧几里得参数空间所带来的挑战。我们的方法利用一种新颖且计算高效的流程来达到更高阶的渐近精度。具体而言，我们开发了一种黎曼流形上的自助法算法，该算法对于假设检验和置信区域构建既计算高效又精确。尽管位置假设检验可以重新表述为一个标准的欧几里得问题，但构建高阶精确的置信区域需要对流形几何进行仔细处理。为此，我们在由二阶回缩诱导的适当坐标表示下建立了高阶渐近理论，从而实现了包含曲率效应的精确展开。我们展示了该框架在各种流形设置下的通用性，包括球面、Stiefel流形、固定秩矩阵流形和秩一张量流形；对于欧几里得子流形，我们还引入了一类具有强一致性性质的类投影坐标图。最后，数值研究证实了所提方法的实际优势。