A Parameter Choice Rule for Tikhonov Regularization Based on Predictive Risk

TL;DR

This paper introduces a new parameter selection rule for Tikhonov regularization that minimizes predictive risk, using an approach based on the signal-to-noise ratio, and demonstrates superior stability and accuracy in simulations.

Contribution

It proposes a novel criterion for regularization parameter choice based on predictive risk minimization that adapts to unknown noise and data norms, improving stability and accuracy.

Findings

Outperforms existing methods like discrepancy principle and GCV in accuracy.

Provides stable solutions for small and large datasets.

Achieves excellent stability and outperforms traditional criteria.

Abstract

In this work, we propose a new criterion for choosing the regularization parameter in Tikhonov regularization when the noise is white Gaussian. The criterion minimizes a lower bound of the predictive risk, when both data norm and noise variance are known, and the parameter choice involves minimizing a function whose solution depends only on the signal-to-noise ratio. Moreover, when neither noise variance nor data norm is given, we propose an iterative algorithm which alternates between a minimization step of finding the regularization parameter and an estimation step of estimating signal-to-noise ratio. Simulation studies on both small- and large-scale datasets suggest that the approach can provide very accurate and stable regularized inverse solutions and, for small sized samples, it outperforms discrepancy principle, balancing principle, unbiased predictive risk estimator, L-curve method generalized cross validation, and quasi-optimality criterion, and achieves excellent stability hitherto unavailable.