Using Supervised Learning to Construct the Best Regularization Term and the Best Multiresolution Analysis

TL;DR

This paper introduces a machine learning approach that jointly learns optimal regularization terms and multiresolution analysis methods for signal processing, connecting harmonic analysis with inverse problems.

Contribution

It proposes a bi-level optimization framework with a convergent sequence of step functions to learn regularization and multiresolution analysis from data.

Findings

Convergence of the proposed step function sequence is proved.

The model effectively learns regularization terms tailored for signal processing.

Connections between wavelet methods, inverse problems, and machine learning are established.

Abstract

By the recent advances in computer technology leading to the invention of more powerful processors, the importance of creating models using data training is even greater than ever. Given the significance of this issue, this work tries to establish a connection among Machine Learning, Inverse Problems, and Applied Harmonic Analysis. Inspired by methods introduced in [12, 17, 22, 30], which are connections between Wavelet and Inverse Problems, we offer a model with the capability of learning in terms of an application in signal processing. In order to reach this model, a bi-level optimization problem will have to be faced. For solving this, a sequence of step functions is presented that its convergence to the solution will be proved. Each of these step functions derives from several constrained optimization problems on $\mathbb{R^n}$ that will be introduced here.