On the convergence of iterative regularization method assisted by the graph Laplacian with early stopping

TL;DR

This paper introduces a novel iterative regularization method that combines classical techniques with a data-driven graph Laplacian, using early stopping based on the discrepancy principle to solve ill-posed inverse problems effectively.

Contribution

The work develops a new iterative scheme integrating a graph Laplacian with early stopping, providing theoretical convergence guarantees and demonstrating robustness across various initial reconstructions.

Findings

The method converges under standard assumptions.

Early stopping via the discrepancy principle ensures stability.

The approach outperforms traditional initializers like FBP, TV, and Tikhonov in experiments.

Abstract

We present a data-assisted iterative regularization method for solving ill-posed inverse problems. The proposed approach, termed \texttt{IRMGL+\(\Psi\)}, integrates classical iterative techniques with a data-driven regularization term realized through an iteratively updated graph Laplacian. Our method commences by computing a preliminary solution using any suitable reconstruction method, which then serves as the basis for constructing the initial graph Laplacian. The solution is subsequently refined through an iterative process, where the graph Laplacian is simultaneously recalibrated at each step to effectively capture the evolving structure of the solution. A key innovation of this work lies in the formulation of this iterative scheme and the rigorous justification of the classical discrepancy principle as a reliable early stopping criterion specifically tailored to the proposed method. Under standard assumptions, we establish stability and convergence results for the scheme when the discrepancy principle is applied. Furthermore, we demonstrate the robustness and effectiveness of our method through numerical experiments utilizing four distinct initial reconstructors $\Psi$: the adjoint operator (Adj), filtered back projection (FBP), total variation (TV) denoising, and standard Tikhonov regularization (Tik). It is observed that \texttt{IRMGL+Adj} demonstrates a distinct advantage over the other initializers, producing a robust and stable approximate solution directly from a basic initial reconstruction.