基于流形几何的降维方法能更好处理弯曲空间数据。
Dimensionality Reduction on Riemannian Manifolds in Data Analysis
- 用测地线距离和切空间表示数据流形结构
- 在球面和对称正定矩阵数据上分类效果优于传统方法
- 适合处理非欧几何数据，如图像、形状分析
本文研究基于黎曼几何的降维方法，旨在保留数据的底层流形结构。重点考察主测地线分析（PGA），作为流形数据的非线性主成分分析推广；并拓展判别分析，通过黎曼适应的其他降维方法实现。这些方法利用测地线距离、切空间表示和内在统计量，获得更忠实的低维嵌入。实验结果表明，在代表性数据集上，黎曼方法相比欧氏方法显著提升表示质量和分类性能，尤其适用于受限于曲面空间（如超球面、对称正定流形）的数据。本研究强调了现代机器学习与数据科学中几何感知降维的重要性。
原文摘要 · Abstract (English)
In this work, we investigate Riemannian geometry based dimensionality reduction methods that respect the underlying manifold structure of the data. In particular, we focus on Principal Geodesic Analysis (PGA) as a nonlinear generalization of PCA for manifold valued data, and extend discriminant analysis through Riemannian adaptations of other known dimensionality reduction methods. These approaches exploit geodesic distances, tangent space representations, and intrinsic statistical measures to achieve more faithful low dimensional embeddings. We also discuss related manifold learning techniques and highlight their theoretical foundations and practical advantages. Experimental results on representative datasets demonstrate that Riemannian methods provide improved representation quality and classification performance compared to their Euclidean counterparts, especially for data constrained to curved spaces such as hyperspheres and symmetric positive definite manifolds. This study underscores the importance of geometry aware dimensionality reduction in modern machine learning and data science applications.
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