A bayesian approach to the estimation of maps between riemannian manifolds, II: examples

TL;DR

This paper applies a Bayesian asymptotic expansion technique to estimate maps between Riemannian manifolds, demonstrating its use through various examples and exploring conditions for constrained regression problems.

Contribution

It extends previous work by applying second-order asymptotic analysis to practical examples and examines first-order conditions for equality-constrained regression on manifolds.

Findings

Derived second-order asymptotic expansion for Bayesian risk in manifold mapping

Applied the technique to diverse examples illustrating its utility

Analyzed first-order conditions for constrained regression problems on manifolds

Abstract

Let M be a smooth compact oriented manifold without boundary, imbedded in a euclidean space E and let f be a smooth map of M into a Riemannian manifold N. An unknown state x in M is observed via X=x+su where s>0 is a small parameter and u is a white Gaussian noise. For a given smooth prior on M and smooth estimators g of the map f we have derived a second-order asymptotic expansion for the related Bayesian risk (see arXiv:0705.2540). In this paper, we apply this technique to a variety of examples. The second part examines the first-order conditions for equality-constrained regression problems. The geometric tools that are utilised in our earlier paper are naturally applicable to these regression problems.