Randomized Projection Methods for Linear Systems with Arbitrarily Large Sparse Corruptions

TL;DR

This paper introduces randomized projection methods to solve large linear systems with sparse, arbitrarily large corruptions, enabling detection of corrupted entries and convergence to the true solution.

Contribution

The paper presents a novel approach that detects corrupted entries in large linear systems and guarantees convergence to the original solution despite corruptions.

Findings

Method effectively detects corrupted entries in large systems.

Approach converges to the true solution despite corruptions.

Experimental results validate the method on real and synthetic data.

Abstract

In applications like medical imaging, error correction, and sensor networks, one needs to solve large-scale linear systems that may be corrupted by a small number of arbitrarily large corruptions. We consider solving such large-scale systems of linear equations $A\mathbf{x}=\mathbf{b}$ that are inconsistent due to corruptions in the measurement vector $\mathbf{b}$. With this as our motivating example, we develop an approach for this setting that allows detection of the corrupted entries and thus convergence to the "true" solution of the original system. We provide analytical justification for our approaches as well as experimental evidence on real and synthetic systems.