Riemannian Principal Component Analysis

TL;DR

This paper introduces Riemannian PCA, a novel framework extending traditional PCA to data on Riemannian manifolds by leveraging local metrics, enabling geometric-aware dimensionality reduction for complex structured datasets.

Contribution

It generalizes Principal Geodesic Analysis to a broader class of datasets using local distance structures, facilitating manifold-aware PCA without relying on global geodesic assumptions.

Findings

Enables PCA on data with manifold structure using local metrics

Provides a unified approach for dimensionality reduction on Riemannian manifolds

Extends PCA applicability to datasets lacking explicit geodesic distances

Abstract

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.