On the performances of a new thresholding procedure using tree structure

TL;DR

This paper introduces a new wavelet thresholding method leveraging tree structures, demonstrating its adaptiveness, near-minimax performance, and superiority over traditional hard thresholding in function estimation within the white noise model.

Contribution

The paper presents a novel wavelet thresholding procedure based on dyadic tree structures, improving adaptiveness and performance over existing methods.

Findings

Achieves near-minimax rates over Besov spaces

Has a large maxiset for convergence rates

Outperforms traditional hard thresholding

Abstract

This paper deals with the problem of function estimation. Using the white noise model setting, we provide a method to construct a new wavelet procedure based on thresholding rules which takes advantage of the dyadic structure of the wavelet decomposition. We prove that this new procedure performs very well since, on the one hand, it is adaptive and near-minimax over a large class of Besov spaces and, on the other hand, the maximal functional space (maxiset) where this procedure attains a given rate of convergence is very large. More than this, by studying the shape of its maxiset, we prove that the new procedure outperforms the hard thresholding procedure.