Principal symmetric space analysis

TL;DR

This paper introduces a new method for principal component analysis on Riemannian symmetric spaces, extending Euclidean PCA to curved manifolds using totally geodesic submanifolds, with applications to various geometric data types.

Contribution

It proposes a novel PCA analogue for Riemannian symmetric spaces utilizing totally geodesic submanifolds, expanding PCA applicability beyond Euclidean spaces.

Findings

Effective PCA approximation on spheres, Grassmannians, tori, and polyspheres.

Demonstrated the method's utility on multiple geometric data types.

Provides a new tool for analyzing data on curved manifolds.

Abstract

We develop a novel analogue of Euclidean PCA (principal component analysis) for data taking values on a Riemannian symmetric space, using totally geodesic submanifolds as approximating lower dimnsional submanifolds. We illustrate the technique on n-spheres, Grassmannians, n-tori and polyspheres.