Flag spaces for Geometric statistics

Flags are sequences of properly embedded linear subspaces. They appear in multiscale dimension reduction methods such as principal or independent component analysis. Non-linear flags also appear to be the right geometric objects to work with for generalizations of PCA to manifolds such as principal nested spheres or barycentric subspace analysis. One can actually show that extracting order principal components actually optimizes a criterion on the flag space and not on Grassmannians as usually thought. Flag spaces are Riemannian homogeneous spaces that generalize Grassmmann and Steifel manifolds. However, they are usually not symmetric. In this talk, I will present an extension of PCA based on flag spaces called Principal subspace analysis that may turn out to be much more stable that the classical PCA decomposition into unidimensional modes. I will also expose a method to obtain confidence regions on the resulting subspaces based on a geometric formulation of the central limit theorem directly in the space of flags. This is joint work with Tom Szwagier for the first part and Dimbihery Rabenoro for the second part.