Barycentric Subspace Analysis: an extension of PCA to Manifolds

This talk address the generalization of Principal Component Analysis
(PCA) to Riemannian manifolds and potentially more general
stratified spaces. Tangent PCA is often sufficient for analyzing
data which are sufficiently centered around a central value
(unimodal or Gaussian-like data), but fails for multimodal or large
support distributions (e.g. uniform on close compact subspaces).
Instead of a covariance matrix analysis, Principal Geodesic Analysis
(PGA) and Geodesic PCA (GPCA) are proposing to minimize the distance
to Geodesic Subspaces (GS) which are spanned by the geodesics going
through a point with tangent vector is a restricted linear sub-space
of the tangent space. Other methods like Principal Nested Spheres
(PNS) restrict to simpler manifolds but emphasize on the need for
the nestedness of the resulting principal subspaces.

We first propose a new and more 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 do not on
tangent vectors, it can also be extended to geodesic spaces which
are not Riemannian. For instance, in stratified spaces, it naturally
allows to have principal subspaces that span over several strata,
which is not the case with PGA. Barycentric subspaces locally
define a submanifold of dimension k which generalizes geodesic
subspaces. Like PGA, barycentric subspaces can naturally be nested,
which allow the construction of inductive forward nested subspaces
approximating data points which contains the Frechet mean. However,
it also allows the construction of backward flags which may not
contain the mean. Second, we rephrase PCA in Euclidean spaces as an
optimization on flags of linear subspaces (a hierarchies of properly
embedded linear subspaces of increasing dimension). We propose for
that an extension of the unexplained variance criterion that
generalizes nicely to flags of barycentric subspaces in Riemannian
manifolds. This results into a particularly appealing generalization
of PCA on manifolds, that we call Barycentric Subspaces Analysis
(BSA). The method will be illustrated on spherical and hyperbolic
spaces, and on diffeomorphisms encoding the deformation of the heart
in cardiac image sequences.