Adaptive Nonparametric Regression on Spin Fiber Bundles

TL;DR

This paper develops an adaptive nonparametric regression method for analyzing functions on spin fiber bundles on the sphere, motivated by astrophysical applications like gravitational lensing, using wavelet thresholding techniques with spin needlets.

Contribution

It extends wavelet thresholding to nonparametric regression on spin fiber bundles, introducing a new adaptive procedure for functions with values as algebraic curves.

Findings

Proposes a thresholding procedure based on spin needlets.

Analyzes rates of convergence and adaptivity over spin Besov spaces.

Applicable to astrophysical data analysis, such as gravitational lensing.

Abstract

The construction of adaptive nonparametric procedures by means of wavelet thresholding techniques is now a classical topic in modern mathematical statistics. In this paper, we extend this framework to the analysis of nonparametric regression on sections of spin fiber bundles defined on the sphere. This can be viewed as a regression problem where the function to be estimated takes as its values algebraic curves (for instance, ellipses) rather than scalars, as usual. The problem is motivated by many important astrophysical applications, concerning for instance the analysis of the weak gravitational lensing effect, i.e. the distortion effect of gravity on the images of distant galaxies. We propose a thresholding procedure based upon the (mixed) spin needlets construction recently advocated by Geller and Marinucci (2008,2010) and Geller et al. (2008,2009), and we investigate their rates of convergence and their adaptive properties over spin Besov balls.