Dimensionality Reduction on Riemannian Manifolds in Data Analysis

TL;DR

This paper explores Riemannian geometry-based dimensionality reduction techniques that preserve the intrinsic structure of manifold-valued data, demonstrating improved data representation and classification performance on curved spaces.

Contribution

It extends PCA to Riemannian manifolds with Principal Geodesic Analysis and adapts discriminant analysis methods, highlighting their advantages over Euclidean approaches.

Findings

Riemannian methods outperform Euclidean counterparts in data representation.

Improved classification accuracy on manifold-constrained datasets.

Effective low-dimensional embeddings for curved space data.

Abstract

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.