Adaptive global thresholding on the sphere

TL;DR

This paper investigates adaptive nonparametric regression estimators on the sphere using needlet-based wavelets, establishing their optimal convergence rates over Besov spaces.

Contribution

It introduces and analyzes the adaptivity and optimality of needlet-based spherical wavelet estimators in nonparametric regression.

Findings

Establishes convergence rates of $L^p$-risks for the estimators.

Proves minimax optimality over Besov spaces.

Demonstrates strong concentration properties of needlets.

Abstract

This work is concerned with the study of the adaptivity properties of nonparametric regression estimators over the $d$-dimensional sphere within the global thresholding framework. The estimators are constructed by means of a form of spherical wavelets, the so-called needlets, which enjoy strong concentration properties in both harmonic and real domains. The author establishes the convergence rates of the $L^p$-risks of these estimators, focussing on their minimax properties and proving their optimality over a scale of nonparametric regularity function spaces, namely, the Besov spaces.