Barycentric Subspace Analysis on Manifolds

TL;DR

This paper generalizes PCA to Riemannian and more general metric spaces using barycentric subspaces, enabling hierarchical and multi-strata analysis of data on complex manifolds.

Contribution

It introduces barycentric subspaces as a new family of subspaces in manifolds, extending PCA beyond Euclidean spaces and allowing analysis on stratified and non-Riemannian spaces.

Findings

Barycentric subspaces are locally submanifolds of dimension k.

Hierarchical nested subspaces can be constructed in manifolds.

The proposed Barycentric Subspaces Analysis generalizes PCA to manifold settings.

Abstract

This paper investigates the generalization of Principal Component Analysis (PCA) to Riemannian manifolds. We first propose a new and general type of family of subspaces in manifolds that we call barycentric subspaces. They are implicitly defined as the locus of points which are weighted means of $k+1$ reference points. As this definition relies on points and not on tangent vectors, it can also be extended to geodesic spaces which are not Riemannian. For instance, in stratified spaces, it naturally allows principal subspaces that span several strata, which is impossible in previous generalizations of PCA. We show that barycentric subspaces locally define a submanifold of dimension k which generalizes geodesic subspaces.Second, we rephrase PCA in Euclidean spaces as an optimization on flags of linear subspaces (a hierarchy of properly embedded linear subspaces of increasing dimension). We show that the Euclidean PCA minimizes the Accumulated Unexplained Variances by all the subspaces of the flag (AUV). Barycentric subspaces are naturally nested, allowing the construction of hierarchically nested subspaces. Optimizing the AUV criterion to optimally approximate data points with flags of affine spans in Riemannian manifolds lead to a particularly appealing generalization of PCA on manifolds called Barycentric Subspaces Analysis (BSA).