2015/12/28 by Stefan Sommer, Sommer, Stefan, Anne Marie Svane +1
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Morphological variations and asymmetry #Statistical and numerical algorithms #Statistics Theory (math.ST)
paper · pdf · doi:10.48550/arxiv.1512.08544
openalex publication_date 2015/12/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We discuss the geometric foundation behind the use of stochastic processes in\nthe frame bundle of a smooth manifold to build stochastic models with\napplications in statistical analysis of non-linear data. The transition\ndensities for the projection to the manifold of Brownian motions developed in\nthe frame bundle lead to a family of probability distributions on the manifold.\nWe explain how data mean and covariance can be interpreted as points in the\nframe bundle or, more precisely, in the bundle of symmetric positive definite\n2-tensors analogously to the parameters describing Euclidean normal\ndistributions. We discuss a factorization of the frame bundle projection map\nthrough this bundle, the natural sub-Riemannian structure of the frame bundle,\nthe effect of holonomy, and the existence of subbundles where the Hormander\ncondition is satisfied such that the Brownian motions have smooth transition\ndensities. We identify the most probable paths for the underlying Euclidean\nBrownian motion and discuss small time asymptotics of the transition densities\non the manifold. The geometric setup yields an intrinsic approach to the\nestimation of mean and covariance in non-linear spaces.\n