A Bayesian approach to the estimation of maps between riemannian manifolds

TL;DR

This paper develops a Bayesian framework for estimating maps between Riemannian manifolds, deriving second-order asymptotic risks and proposing minimax estimators based on harmonic map theory.

Contribution

It introduces a second-order asymptotic analysis for Bayesian risk in manifold mapping and constructs minimax estimators using harmonic maps and hypo-elliptic operators.

Findings

Derived second-order asymptotic expansion for Bayesian risk.

Proposed a second-order minimax estimator for the map.

Utilized geometry of manifolds and harmonic maps in estimation.

Abstract

Let \Theta be a smooth compact oriented manifold without boundary, embedded in a euclidean space and let \gamma be a smooth map \Theta into a riemannian manifold \Lambda. An unknown state \theta \in \Theta is observed via X=\theta+\epsilon \xi where \epsilon>0 is a small parameter and \xi is a white Gaussian noise. For a given smooth prior on \Theta and smooth estimator g of the map \gamma we derive a second-order asymptotic expansion for the related Bayesian risk. The calculation involves the geometry of the underlying spaces \Theta and \Lambda, in particular, the integration-by-parts formula. Using this result, a second-order minimax estimator of \gamma is found based on the modern theory of harmonic maps and hypo-elliptic differential operators.