{"paper":{"title":"On the interval of fluctuation of the singular values of random matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.IT","math.FA","math.IT"],"primary_cat":"math.PR","authors_text":"Alain Pajor, Alexander E. Litvak, Nicole Tomczak-Jaegermann, Olivier Gu\\'edon","submitted_at":"2015-09-08T11:05:17Z","abstract_excerpt":"Let $A$ be a matrix whose columns $X_1,\\dots, X_N$ are independent random vectors in $\\mathbb{R}^n$. Assume that the tails of the 1-dimensional marginals decay as $\\mathbb{P}(|\\langle X_i, a\\rangle|\\geq t)\\leq t^{-p}$ uniformly in $a\\in S^{n-1}$ and $i\\leq N$. Then for $p>4$ we prove that with high probability $A/{\\sqrt{n}}$ has the Restricted Isometry Property (RIP) provided that Euclidean norms $|X_i|$ are concentrated around $\\sqrt{n}$. We also show that the covariance matrix is well approximated by the empirical covariance matrix and establish corresponding quantitative estimates on the ra"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.02322","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"}