Sample-Limited $L_p$ Barycentric Subspace Analysis on Constant Curvature Spaces

Abstract

Generalizing Principal Component Analysis (PCA) to manifolds is pivotal for many statistical applications on geometric data. We rely in this paper on barycentric subspaces, implicitly defined as the locus of points which are weighted means of $k+1$ reference points [8, 9]. Barycentric subspaces can naturally be nested and allow the construction of inductive forward or backward nested subspaces approximating data points. We can also consider the whole hierarchy of embedded barycentric subspaces defined by an ordered series of points in the manifold (a flag of affine spans): optimizing the accumulated unexplained variance (AUV) over all the subspaces actually generalizes PCA to non Euclidean spaces, a procedure named Barycentric Subspaces Analysis (BSA).

In this paper, we first investigate sample-limited inference algorithms where the optimization is limited to the actual data points: this transforms a general optimization into a simple enumeration problem. Second, we propose to robustify the criterion by considering the unexplained p-variance of the residuals instead of the classical 2-variance. This construction is very natural with barycentric subspaces since the affine span is stable under the choice of the value of p. The proposed algorithms are illustrated on examples in constant curvature spaces: optimizing the (accumulated) unexplained p-variance ($L_p$ PBS and BSA) for $0<p \le 1$ can identify reference points in clusters of a few points within a large number of random points in spheres and hyperbolic spaces.