{"paper":{"title":"On the Optimal Weighted $\\ell_2$ Regularization in Overparameterized Linear Regression","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.LG","math.ST","stat.TH"],"primary_cat":"stat.ML","authors_text":"Denny Wu, Ji Xu","submitted_at":"2020-06-10T12:38:43Z","abstract_excerpt":"We consider the linear model $\\mathbf{y} = \\mathbf{X} \\mathbf{\\beta}_\\star + \\mathbf{\\epsilon}$ with $\\mathbf{X}\\in \\mathbb{R}^{n\\times p}$ in the overparameterized regime $p>n$. We estimate $\\mathbf{\\beta}_\\star$ via generalized (weighted) ridge regression: $\\hat{\\mathbf{\\beta}}_\\lambda = \\left(\\mathbf{X}^T\\mathbf{X} + \\lambda \\mathbf{\\Sigma}_w\\right)^\\dagger \\mathbf{X}^T\\mathbf{y}$, where $\\mathbf{\\Sigma}_w$ is the weighting matrix. Under a random design setting with general data covariance $\\mathbf{\\Sigma}_x$ and anisotropic prior on the true coefficients $\\mathbb{E}\\mathbf{\\beta}_\\star\\mat"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2006.05800","kind":"arxiv","version":4},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2006.05800/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}