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

TL;DR

The paper tackles ill-posed linear inverse problems by introducing IRMGL+$\Psi$, an iterative regularization method that incorporates a data-driven graph Laplacian updated at every iteration. By starting from a preliminary reconstruction $\Psi(v^\delta)$ and performing updates $u_{k+1}^\delta = u_k^\delta - \alpha_k^\delta A^*(Au_k^\delta - v^\delta) - \beta_k^\delta \Delta_{u_k^\delta}u_k^\delta$, the approach blends classical data fidelity with a learned, evolving regularizer. The authors develop rigorous convergence and stability results under the discrepancy principle, and validate the method through CT and image deblurring experiments, showing particularly strong performance with the Adj initial reconstructer. Compared with static and iteratively updated graph-Laplacian variational methods (GraphLa+$\Psi$ and it-GraphLa+$\Psi$), IRMGL+$\Psi$ offers a scalable, regularization-by-early-stopping framework with competitive reconstruction quality and favorable computational efficiency. The work advances data-driven regularization in inverse problems by delivering both theoretical guarantees and practical algorithms that adapt graph structure during reconstruction.

Abstract

We present a data-assisted iterative regularization method for solving ill-posed inverse problems. The proposed approach, termed \texttt{IRMGL+\(Ψ\)}, 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 $Ψ$: 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.

Figures (12)

• Figure 1: An abstract illustration of the IRMGL+$\Psi$ method. The initial reconstructor $\Psi$ does not necessarily serve as a regularization method, as indicated by the piecewise linear trajectory of $\Psi(v^\delta)$ as $\delta \to 0$. However, when combined with IRMGL+$\Psi$, the resulting process constitutes a stable and convergent regularization method. The dotted trajectory represents the iterative progression of the method, while the approximate solution at iteration $k_\delta$ is denoted by $u_{k_\delta}^\delta$. The coiled path illustrates the regularization behavior of the method, highlighting the convergence $u_{k_\delta}^\delta \to u$ as $\delta \to 0$. Full mathematical justification for the convergence is given in subsequent sections. The difference between IRMGL+$\Psi$ and GraphLa+$\Psi$ can be noted using bianchi2025data.
• Figure 1: $2\times2$ grayscale image with an induced graph. Each node corresponds to a pixel and the red edge labels indicate the weights based on similarity and spatial proximity.
• Figure 1: True image in grayscale.
• Figure 1: Left: True phantom, Right: Noisy sinogram $v^\delta$ with $\delta = 0.05$.
• Figure 2: This figure provides a step-by-step illustration of the process of converting an image $\mathbf{x}$ into a graph representation. On left, a $16 \times 16$ grayscale image is displayed, where the intensity of each pixel is determined by the function $\mathbf{x}$, which encapsulates the underlying image information. In second panel, each pixel is mapped to a graph node (depicted as a sky blue circle), with its location corresponding to its discrete position in the grid $\mathbb{Z}^2$. Edges between nodes are then constructed using the data-dependent weight function $w_{\mathbf{x}}(a, b)$, parameterized by $R = 1$ and $\sigma = 0.005$, which quantifies similarity based on pixel intensities. The strength of these connections is visually represented by the thickness of red edges: thicker edges indicate higher similarity between adjacent pixels, while thinner edges reflect greater dissimilarity. The right panel illustrates the effect of increasing the neighborhood radius to $R = 2$, resulting in a denser connectivity pattern that incorporates a broader local context.

Theorems & Definitions (20)

• Definition 2.1
• Example 2.3
• Remark 3.3
• Remark 3.4
• Remark 3.5
• Lemma 3.6
• Proof 1
• Theorem 3.7
• Proof 2
• Lemma 3.8