Talk by Xavier Pennec
Geometric Statistics beyond the Riemannian structure and beyond the mean
Geometric statistics aims at defining and studying statistics on objects with a geometric structure. Statistics on non-linear spaces are now well developed in Riemannian manifolds and metric spaces. Their generalization to other geometric structures such as affine connection spaces turn out to be important for Lie groups, which are naturally endows with a globally affine symmetric space structure thanks to the symmetry s_p(q) = p q^-1 p. The resulting (generally non-metric) Cartan Schouten connection provides strong theoretical bases for the use of one-parameter subgroups and allows us to define bi-invariant means on Lie groups even in the absence of a bi-invariant metric. It was recently realized that the symmetric space structure also notably simplifies some algorithms. For instance, parallel transport can be achieved exactly with transvections (one step of Pole ladder).
In order to define interesting summary statistics of geometric data, one also often has to go beyond the 0-dimensional approximation of their mean. A recent advance in this direction focuses on flags (sequences of properly nested) of affine spans for generalizing PCA to these Riemannian and affine manifolds. Barycentric subspaces and affine spans, defined as the (completion of the) locus of weighted means to a number of reference points, can be naturally nested by defining an ordering of the reference points. Like for PGA, this allows the construction of forward or backward nested sequence of subspaces. However, these methods optimized for one subspace at a time and cannot optimize the unexplained variance simultaneously for all the subspaces of the flag.
In order to obtain a global criterion, PCA in Euclidean spaces is rephrased as an optimization on the flags of linear subspaces and we propose an extension of the unexplained variance criterion that generalizes nicely to flags of affine spans in Riemannian manifolds.
This results into a particularly appealing generalization of PCA on manifolds, that was 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.