Flexible Krylov methods for $\ell_p$ regularization

TL;DR

This paper introduces flexible Krylov methods for large-scale $ ext{ell}_p$ regularization problems, improving efficiency and avoiding complex parameter tuning through adaptive reweighted norms and flexible preconditioning.

Contribution

It develops a novel flexible Krylov framework for $ ext{ell}_p$ regularization, including a flexible Golub-Kahan approach, enhancing computational efficiency and solution sparsity.

Findings

Numerical results show improved image deblurring and tomography reconstruction.

The method efficiently computes sparse solutions for $p=1$ cases.

Avoids inner-outer schemes and complex regularization parameter tuning.

Abstract

In this paper we develop flexible Krylov methods for efficiently computing regularized solutions to large-scale linear inverse problems with an $\ell_2$ fit-to-data term and an $\ell_p$ penalization term, for $p\geq 1$. First we approximate the $p$-norm penalization term as a sequence of $2$-norm penalization terms using adaptive regularization matrices in an iterative reweighted norm fashion, and then we exploit flexible preconditioning techniques to efficiently incorporate the weight updates. To handle general (non-square) $\ell_p$-regularized least-squares problems, we introduce a flexible Golub-Kahan approach and exploit it within a Krylov-Tikhonov hybrid framework. The key benefits of our approach compared to existing optimization methods for $\ell_p$ regularization are that efficient projection methods replace inner-outer schemes and that expensive regularization parameter selection techniques can be avoided. Theoretical insights are provided, and numerical results from image deblurring and tomographic reconstruction illustrate the benefits of this approach, compared to well-established methods. Furthermore, we show that our approach for $p=1$ can be used to efficiently compute solutions that are sparse with respect to some transformations.