Notes de dernières minutes : Cônes dans les K3
Résumé : At the interface of geometry, statistics, image analysis and medicine, computational anatomy aims at learning the biological variability of the organs shapes and their dynamics at the population level. Since shapes and deformations live in non-linear spaces, this requires a consistent statistical framework on manifolds and Lie groups, which has motivated the development of Geometric Statistics during the last decade. I address in this talk the generalization of Principal Component Analysis (PCA) to Riemannian manifolds and potentially more general stratified spaces. The classical tangent PCA developing data in the tangent space at the mean often fails for multimodal or large support distributions. Other techniques like Principal Geodesic Analysis (PGA) and Geodesic PCA (GPCA) that minimize the distance to Geodesic Subspaces (GS), are still based on subspaces defined by a central point or a central geodesic. We first propose Barycentric Subspaces (BS), a new and more general type of family of subspaces in manifolds, implicitly defined as the locus of weighted means of k+1 reference points. BS locally define a submanifold of dimension k which generalizes geodesic subspaces. Barycentric subspaces can naturally be nested, which allow the construction of inductive forward or backward nested subspaces approximating data points. In order to be optimal for all dimensions together, we rephrase PCA in Euclidean spaces as an optimization on flags of linear subspaces (a hierarchies of properly embedded linear subspaces of increasing dimension) using 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 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.
Lieu : Bâtiment 1R3, salle de conférence MIP (1er étage)
Résumé : We present a tensor representation for spin states that finds applications in the characterization of the convex hull of spin coherent states. By making a connection between this problem an the truncated moment sequence problem, we develop an algorithm for the detection of entanglement in symmetric states.
Lieu : Salle de Séminaire IRSAMC, portes N.325 et 327- 3e étage