Adaptive Bregman-Kaczmarz: An Approach to Solve Linear Inverse Problems with Independent Noise Exactly

TL;DR

This paper introduces an adaptive block Bregman-Kaczmarz method for linear inverse problems with independent noise, demonstrating convergence to the exact solution using data-driven stepsize estimation.

Contribution

It proposes a novel adaptive stepsize strategy for the block Bregman-Kaczmarz method under independent noise, with practical data-based estimation techniques.

Findings

Convergence to the exact solution with the adaptive stepsize.

Effective heuristic estimates for stepsize calculation.

Numerical experiments confirm the method's practicality.

Abstract

We consider the block Bregman-Kaczmarz method for finite dimensional linear inverse problems. The block Bregman-Kaczmarz method uses blocks of the linear system and performs iterative steps with these blocks only. We assume a noise model that we call independent noise, i.e. each time the method performs a step for some block, one obtains a noisy sample of the respective part of the right-hand side which is contaminated with new noise that is independent of all previous steps of the method. One can view these noise models as making a fresh noisy measurement of the respective block each time it is used. In this framework, we are able to show that a well-chosen adaptive stepsize of the block Bergman-Kaczmarz method is able to converge to the exact solution of the linear inverse problem. The plain form of this adaptive stepsize relies on unknown quantities (like the Bregman distance to the solution), but we show a way how these quantities can be estimated purely from given data. We illustrate the finding in numerical experiments and confirm that these heuristic estimates lead to effective stepsizes.