Abstract - Geometric Statistics on Manifolds and Shape Spaces.
- Stéphanie Allasonnière (School of medicine, Paris Descartes university, France)
- Xavier Pennec (INRIA sophia, France)
- Marc Arnaudon (IMB, France)
- Aasa Feragen (DIKU, Denmark)
- Stanley Durrleman (ARAMIS lab, France)
- Ian Dryden (University of Nottingham, UK)
- Alice Le Brigant (IMB, Université de Bordeaux, France)
This session presents recent progresses in geometric statistics. In many applications domains such as computational anatomy, computer vision, structural biology, computational phylogenetics, etc one models data as elements of a manifold which is quotiented by a proper and isometric Lie group action (a shape space). For instance, Kendall shape spaces encode the invariants of the configuration of a fixed number of points under the action of Euclidean or similarity transformations. One can also construct (infinite dimensional) spaces of curves or surfaces by quotienting out the parametrization. Shape spaces are stratified spaces whose manifold part is Riemannian. Looking for summary, explanatory or discriminative statistics on such measurements is first complicated by the presence of the curvature. The theory of statistics on Riemannian manifolds now begins to be powerful enough to provide simple notions (mean values, moments) and some (limited) central limit theorems, although the role of the curvature is not fully elucidated. However, subspace estimation methods going beyond the mean value have only recently emerged. A second complication arise in quotient spaces when data are scattered over several regular strata or close to singular points of the stratification. Other differential geometric structures such as affine connection spaces or fiber spaces raise new questions for the generalization of statistical notions and estimation methods.