Weighted Thresholding and Nonlinear Approximation

TL;DR

This paper introduces a weighted thresholding approach for nonlinear approximation using redundant dictionaries, providing theoretical error bounds and demonstrating its effectiveness in denoising music signals compared to existing methods.

Contribution

It proposes a novel weighted thresholding method that accounts for coefficient dependencies, with proven error bounds and practical comparison to other denoising techniques.

Findings

The method achieves lower reconstruction error than traditional greedy algorithms.

It provides a strong Jackson embedding with explicit error bounds.

Demonstrates improved denoising performance on music signals using Gabor dictionaries.

Abstract

We present a new method for performing nonlinear approximation with redundant dictionaries. The method constructs an $m-$term approximation of the signal by thresholding with respect to a weighted version of its canonical expansion coefficients, thereby accounting for dependency between the coefficients. The main result is an associated strong Jackson embedding, which provides an upper bound on the corresponding reconstruction error. To complement the theoretical results, we compare the proposed method to the pure greedy method and the Windowed-Group Lasso by denoising music signals with elements from a Gabor dictionary.