{"paper":{"title":"Minimax rates of estimation for high-dimensional linear regression over $\\ell_q$-balls","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.IT","math.IT","stat.TH"],"primary_cat":"math.ST","authors_text":"Bin Yu, Garvesh Raskutti, Martin J. Wainwright","submitted_at":"2009-10-11T20:38:57Z","abstract_excerpt":"Consider the standard linear regression model $\\y = \\Xmat \\betastar + w$, where $\\y \\in \\real^\\numobs$ is an observation vector, $\\Xmat \\in \\real^{\\numobs \\times \\pdim}$ is a design matrix, $\\betastar \\in \\real^\\pdim$ is the unknown regression vector, and $w \\sim \\mathcal{N}(0, \\sigma^2 I)$ is additive Gaussian noise. This paper studies the minimax rates of convergence for estimation of $\\betastar$ for $\\ell_\\rpar$-losses and in the $\\ell_2$-prediction loss, assuming that $\\betastar$ belongs to an $\\ell_{\\qpar}$-ball $\\Ballq(\\myrad)$ for some $\\qpar \\in [0,1]$. We show that under suitable regu"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0910.2042","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"}