The talk presents the general setting of geometric stastics (statistics on objects with a gemetric structure) with and emphasis on the Riemannian and affine connection space. A recent advance in this field focuses on flags (sequences of properly nested) of affine spans for generalizing PCA to Riemannian 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 we call Barycentric Subspaces Analysis (BSA). The second part of the talk detail and partially extend the statistical Riemannian framework to affine connection spaces and more particularly to Lie groups provided with the canonical Cartan-Schouten connection (a non-metric connection). This provides strong theoretical bases for the use of one-parameter subgroups and allows to define bi-invariant means on Lie groups even in the absence of a bi-invariant metric which is an important achievement since bi-invariant metrics only exists on the direct product of compact and abelian groups). Lie groups can also be considered as globally affine symmetric space structure thanks to the symmetry s_p(q) = p q^-1 p. The symmetric Cartan-Schouten connection turn also to be the cannonical connection associated with this symmetric structure. This notably simplifies some algorithms like parallel transport as we can prove that the Pole ladder previously introduced is actually an exact scheme in one step only in symmetric spaces.
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