{"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/group-regularized-ridge-regression-via","title":"$σ$-Ridge: group regularized ridge regression via empirical Bayes noise level cross-validation","arxiv_id":"2010.15817","date":"2020-10-29","proceeding":null,"authors":["Nikolaos Ignatiadis","Panagiotis Lolas"],"abstract":"Features in predictive models are not exchangeable, yet common supervised models treat them as such. Here we study ridge regression when the analyst can partition the features into $K$ groups based on external side-information. For example, in high-throughput biology, features may represent gene expression, protein abundance or clinical data and so each feature group represents a distinct modality. The analyst's goal is to choose optimal regularization parameters $\\lambda = (\\lambda_1, \\dotsc, \\lambda_K)$ -- one for each group. In this work, we study the impact of $\\lambda$ on the predictive risk of group-regularized ridge regression by deriving limiting risk formulae under a high-dimensional random effects model with $p\\asymp n$ as $n \\to \\infty$. Furthermore, we propose a data-driven method for choosing $\\lambda$ that attains the optimal asymptotic risk: The key idea is to interpret the residual noise variance $\\sigma^2$, as a regularization parameter to be chosen through cross-validation. An empirical Bayes construction maps the one-dimensional parameter $\\sigma$ to the $K$-dimensional vector of regularization parameters, i.e., $\\sigma \\mapsto \\widehat{\\lambda}(\\sigma)$. Beyond its theoretical optimality, the proposed method is practical and runs as fast as cross-validated ridge regression without feature groups ($K=1$).","url_abs":"https://arxiv.org/abs/2010.15817v2","url_pdf":"https://arxiv.org/pdf/2010.15817v2.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":"group-regularized-ridge-regression-via","repo_url":"https://github.com/nignatiadis/SigmaRidgeRegression.jl","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"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}