Barycentric Subspaces Analysis on Spheres

This paper addresses the generalization of Principal Component Analysis (PCA) to Riemannian manifolds. Current methods like Principal Geodesic Analysis (PGA) and Geodesic PCA (GPCA) minimize the distance to a ”Geodesic subspace”. This allows to build sequences of nested subspaces which are consistent with a forward component analysis approach. However, these methods cannot easily be adapted to a backward analysis and alck symmetric in the parametrization of the subspaces. We propose in [10] a new and more general type of family of subspaces in manifolds, barycentric subspaces, which are implicitly defined as the locus of points which are weighted means of k + 1 reference points. Depending on the generalization of the mean that we use, we obtain the Fréchet / Karcher / Exponential barycentric subspaces (FBS / KBS / EBS). The completion of the last one is called the affine span. These definitions were shown to define locally submanifolds of dimension k. In this paper, we investigate barycentric subspaces in one of the simplest manifold: the sphere. We show that the affine span is a great subsphere in generic conditions, i.e. also a geodesic subspace. This coincidence of spaces is due to the very high symmetry of the sphere. For second order jets, we show that we obtain subspheres of different radii as in the case of principal nested spheres (PNS) analysis. Among the points of the affine span, determining which ones belong to the Karcher barycentric subspaces (KBS) turns out to be a surprisingly difficult problem. Practical experiments show that the KBS covers in general only a small portion of the subsphere containing the reference points. This suggests that the affine span might be a much more interesting definition to work with for subspace definition purposes. We finally discuss the use to these barycentric subspaces to generalize PCA on manifolds, a procedure that we name Barycentric Subpsace Analysis (BSA). Like PGA, barycentric subspaces allow the construction of a forward nested sequence of subspaces which contains the Fréchet mean. However, the definition also allows the construction of backward nested sequence which may not contain the mean.