$\ell^1$ penalty for ill-posed inverse problems

TL;DR

This paper investigates the use of an $\ell^1$ penalty in ill-posed inverse problems, demonstrating that with an appropriate loss function, it yields adaptive estimators that achieve optimal convergence rates without prior regularity knowledge.

Contribution

It introduces a method using $\ell^1$ penalty with a suitable loss function to produce adaptive estimators for ill-posed inverse problems, achieving optimal convergence rates.

Findings

Adaptive estimators converge at optimal rates

Proper loss function selection is crucial

Method applies to numerical analysis and image deblurring

Abstract

We tackle the problem of recovering an unknown signal observed in an ill-posed inverse problem framework. More precisely, we study a procedure commonly used in numerical analysis or image deblurring: minimizing an empirical loss function balanced by an $l^1$ penalty, acting as a sparsity constraint. We prove that, by choosing a proper loss function, this estimation technique enables to build an adaptive estimator, in the sense that it converges at the optimal rate of convergence without prior knowledge of the regularity of the true solution