Manifold Stability for Means of Random Elements on Quotients Due to Isometric Lie Group Actions

• Activity Number: 605
• Type: Topic Contributed
• Date/Time: Thursday, August 4, 2011 : 8:30 AM to 10:20 AM
• Sponsor: IMS
• Author(s): Stephan Huckemann*+
• Companies: Institut fur Mathematische Stochastik
• Keywords: Shape Analysis ; Riemannian Manifold ; Frechet Mean ; Stratified Space ; Intrinsic Mean ; Central Limit Theorem

Abstract

Quotients due to compact Lie group acting isometrically on a Riemannian manifold are usually no longer Riemannian manifolds but stratifications of Riemannian manifolds of different dimensions. In view of a Central-Limit-Theorem one would like to rule out that a mean of a random element is assumed on a lower dimensional manifold stratum. We show that this is the case for intrinsic means as well as for Ziezold means if the action is isometric with respect to the underlying extrinsic distance. For similarly obtained 3D Procrustes means, a counterexample is given.