Optimal regularized inverse matrices for inverse problems

TL;DR

This paper introduces a method for efficiently updating regularized inverse matrices with low-rank modifications, improving solutions for large-scale inverse problems like image deblurring and tomography.

Contribution

It provides an explicit solution for rank-constrained regularized inverse approximations allowing updates and probabilistic information integration, with an efficient rank-update algorithm.

Findings

Rank-updates improve inverse problem accuracy

Method effective for large, sparse matrices

Enhanced solutions in image deblurring and tomography

Abstract

In this paper, we consider optimal low-rank regularized inverse matrix approximations and their applications to inverse problems. We give an explicit solution to a generalized rank-constrained regularized inverse approximation problem, where the key novelties are that we allow for updates to existing approximations and we can incorporate additional probability distribution information. Since computing optimal regularized inverse matrices under rank constraints can be challenging, especially for problems where matrices are large and sparse or are only accessable via function call, we propose an efficient rank-update approach that decomposes the problem into a sequence of smaller rank problems. Using examples from image deblurring, we demonstrate that more accurate solutions to inverse problems can be achieved by using rank-updates to existing regularized inverse approximations. Furthermore, we show the potential benefits of using optimal regularized inverse matrix updates for solving perturbed tomographic reconstruction problems.