QuantileRK: Solving Large-Scale Linear Systems with Corrupted, Noisy Data

TL;DR

QuantileRK is an iterative method designed to efficiently solve large-scale linear systems affected by both sparse corruptions and widespread noise, with proven exponential convergence and robustness.

Contribution

This paper extends the analysis of QuantileRK to systems with both corruptions and noise, demonstrating its convergence and robustness in more realistic scenarios.

Findings

QuantileRK converges exponentially for systems with corruptions and noise.

Theoretical analysis confirms convergence rate up to an error threshold.

Experimental results validate QuantileRK's effectiveness in noisy, corrupted data environments.

Abstract

Measurement data in linear systems arising from real-world applications often suffers from both large, sparse corruptions, and widespread small-scale noise. This can render many popular solvers ineffective, as the least squares solution is far from the desired solution, and the underlying consistent system becomes harder to identify and solve. QuantileRK is a member of the Kaczmarz family of iterative projective methods that has been shown to converge exponentially for systems with arbitrarily large sparse corruptions. In this paper, we extend the analysis to the case where there are not only corruptions present, but also noise that may affect every data point, and prove that QuantileRK converges with the same rate up to an error threshold. We give both theoretical and experimental results demonstrating QuantileRK's strength.