{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/a-data-dependent-regularization-method-based","title":"A data-dependent regularization method based on the graph Laplacian","arxiv_id":"2312.16936","date":"2023-12-28","proceeding":null,"authors":["Davide Bianchi","Davide Evangelista","Stefano Aleotti","Marco Donatelli","Elena Loli Piccolomini","Wenbin Li"],"abstract":"We investigate a variational method for ill-posed problems, named $\\texttt{graphLa+}\\Psi$, which embeds a graph Laplacian operator in the regularization term. The novelty of this method lies in constructing the graph Laplacian based on a preliminary approximation of the solution, which is obtained using any existing reconstruction method $\\Psi$ from the literature. As a result, the regularization term is both dependent on and adaptive to the observed data and noise. We demonstrate that $\\texttt{graphLa+}\\Psi$ is a regularization method and rigorously establish both its convergence and stability properties. We present selected numerical experiments in 2D computerized tomography, wherein we integrate the $\\texttt{graphLa+}\\Psi$ method with various reconstruction techniques $\\Psi$, including Filter Back Projection ($\\texttt{graphLa+FBP}$), standard Tikhonov ($\\texttt{graphLa+Tik}$), Total Variation ($\\texttt{graphLa+TV}$), and a trained deep neural network ($\\texttt{graphLa+Net}$). The $\\texttt{graphLa+}\\Psi$ approach significantly enhances the quality of the approximated solutions for each method $\\Psi$. Notably, $\\texttt{graphLa+Net}$ is outperforming, offering a robust and stable application of deep neural networks in solving inverse problems.","url_abs":"https://arxiv.org/abs/2312.16936v2","url_pdf":"https://arxiv.org/pdf/2312.16936v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"a-data-dependent-regularization-method-based","repo_url":"https://github.com/devangelista2/graphlaplus","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}