{"paper":{"title":"On higher order isotropy conditions and lower bounds for sparse quadratic forms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["stat.TH"],"primary_cat":"math.ST","authors_text":"Alan Muro, Sara van de Geer","submitted_at":"2014-05-23T09:13:23Z","abstract_excerpt":"This study aims at contributing to lower bounds for empirical compatibility constants or empirical restricted eigenvalues. This is of importance in compressed sensing and theory for $\\ell_1$-regularized estimators. Let $X$ be an $n \\times p$ data matrix with rows being independent copies of a $p$-dimensional random variable. Let $\\hat \\Sigma := X^T X / n$ be the inner product matrix. We show that the quadratic forms $u^T \\hat \\Sigma u$ are lower bounded by a value converging to one, uniformly over the set of vectors $u$ with $u^T \\Sigma_0 u $ equal to one and $\\ell_1$-norm at most $M$. Here $\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1405.5995","kind":"arxiv","version":2},"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"}