Perturbation-Based Regularization for Signal Estimation in Linear Discrete Ill-posed Problems

TL;DR

This paper introduces a novel regularization method for linear discrete ill-posed problems that improves solution stability by enhancing singular-value structure, optimizing mean-squared error, and outperforming existing methods in accuracy, robustness, and runtime.

Contribution

A new regularization technique based on singular-value enhancement and a novel parameter selection method for better solutions in ill-posed problems.

Findings

Outperforms benchmark regularization methods in accuracy.

Offers the lowest runtime among tested methods.

Provides the highest robustness in diverse scenarios.

Abstract

Estimating the values of unknown parameters from corrupted measured data faces a lot of challenges in ill-posed problems. In such problems, many fundamental estimation methods fail to provide a meaningful stabilized solution. In this work, we propose a new regularization approach and a new regularization parameter selection approach for linear least-squares discrete ill-posed problems. The proposed approach is based on enhancing the singular-value structure of the ill-posed model matrix to acquire a better solution. Unlike many other regularization algorithms that seek to minimize the estimated data error, the proposed approach is developed to minimize the mean-squared error of the estimator which is the objective in many typical estimation scenarios. The performance of the proposed approach is demonstrated by applying it to a large set of real-world discrete ill-posed problems. Simulation results demonstrate that the proposed approach outperforms a set of benchmark regularization methods in most cases. In addition, the approach also enjoys the lowest runtime and offers the highest level of robustness amongst all the tested benchmark regularization methods.