Priorconditioned Sparsity-Promoting Projection Methods for Deterministic and Bayesian Linear Inverse Problems

TL;DR

This paper introduces priorconditioned sparsity-promoting projection methods, enhancing convergence and efficiency in solving high-dimensional inverse problems with sharp edge signals, by integrating priorconditioning into Krylov subspace techniques.

Contribution

It proposes the PS-GKS method that combines priorconditioning with GKS to improve convergence and automatic hyper-parameter selection in sparsity-promoting inverse problems.

Findings

PS-GKS outperforms existing hybrid Krylov methods in numerical tests.

Priorconditioning accelerates convergence in sparsity-promoting inverse problems.

Recycling and restarting variants effectively handle memory limitations.

Abstract

High-quality reconstructions of signals and images with sharp edges are needed in a wide range of applications. To overcome the large dimensionality of the parameter space and the complexity of the regularization functional, {sparisty-promoting} techniques for both deterministic and hierarchical Bayesian regularization rely on solving a sequence of high-dimensional iteratively reweighted least squares (IRLS) problems on a lower-dimensional subspace. Generalized Krylov subspace (GKS) methods are a particularly potent class of hybrid Krylov schemes that efficiently solve sequences of IRLS problems by projecting large-scale problems into a relatively small subspace and successively enlarging it. We refer to methods that promote sparsity and use GKS as S-GKS. A disadvantage of S-GKS methods is their slow convergence. In this work, we propose techniques that improve the convergence of S-GKS methods by combining them with priorconditioning, which we refer to as PS-GKS. Specifically, integrating the PS-GKS method into the IAS algorithm allows us to automatically select the shape/rate parameter of the involved generalized gamma hyper-prior, which is often fine-tuned otherwise. Furthermore, we proposed and investigated variations of the proposed PS-GKS method, including restarting and recycling (resPS-GKS and recPS-GKS). These respectively leverage restarted and recycled subspaces to overcome situations when memory limitations of storing the basis vectors are a concern. We provide a thorough theoretical analysis showing the benefits of priorconditioning for sparsity-promoting inverse problems. Numerical experiment are used to illustrate that the proposed PS-GKS method and its variants are competitive with or outperform other existing hybrid Krylov methods.