标题： 黎曼主成分分析 显示英文标题

标题： Riemannian Principal Component Analysis

摘要： 本文提出了一种创新的主成分分析（PCA）扩展方法，超越了传统假设数据位于欧几里得空间的限制，使其能够应用于黎曼流形上的数据。 主要解决的挑战是在这些流形上缺乏向量空间运算。 Fletcher等人在其工作{\em 非线性形状统计的主测地线分析}中提出了主测地线分析（PGA）作为一种几何方法来分析黎曼流形上的数据，尤其适用于像医学图像这样结构化的数据集，在这些数据集中流形的内在结构显而易见。 然而，当处理缺乏隐式局部距离概念的一般数据集时，PGA的应用受到限制。 在这项工作中，我们引入了一个广义框架，称为{\em 黎曼主成分分析（R-PCA）}，以扩展PGA适用于任何具有局部距离结构的数据。 具体来说，我们通过为数据表配备局部度量来适应PCA方法到黎曼流形，从而实现流形几何的融合。 此框架提供了一种统一的方法，在流形上直接进行降维和统计分析，为具有区域特定或部分特定距离概念的数据集打开了新的可能性，确保尊重它们的内在几何属性。 显示英文摘要

摘要： This paper proposes an innovative extension of Principal Component Analysis (PCA) that transcends the traditional assumption of data lying in Euclidean space, enabling its application to data on Riemannian manifolds. The primary challenge addressed is the lack of vector space operations on such manifolds. Fletcher et al., in their work {\em Principal Geodesic Analysis for the Study of Nonlinear Statistics of Shape}, proposed Principal Geodesic Analysis (PGA) as a geometric approach to analyze data on Riemannian manifolds, particularly effective for structured datasets like medical images, where the manifold's intrinsic structure is apparent. However, PGA's applicability is limited when dealing with general datasets that lack an implicit local distance notion. In this work, we introduce a generalized framework, termed {\em Riemannian Principal Component Analysis (R-PCA)}, to extend PGA for any data endowed with a local distance structure. Specifically, we adapt the PCA methodology to Riemannian manifolds by equipping data tables with local metrics, enabling the incorporation of manifold geometry. This framework provides a unified approach for dimensionality reduction and statistical analysis directly on manifolds, opening new possibilities for datasets with region-specific or part-specific distance notions, ensuring respect for their intrinsic geometric properties.