Approximation of Densities on Riemannian Manifolds

Abstract

Finding an approximate probability distribution best representing a sample on a measure space is one of the most basic operations in statistics. Many procedures were designed for that purpose when the underlying space is a finite dimensional Euclidean space. In applications, however, such a simple setting may not be adapted and one has to consider data living on a Riemannian manifold. The lack of unique generalizations of the classical distributions, along with theoretical and numerical obstructions require several options to be considered. The present work surveys some possible extensions of well known families of densities to the Riemannian setting, both for parametric and non-parametric estimation.

Keywords

• quantization
• directional densities
• exponential family
• group invariance
• Riemannian manifold