{"paper":{"title":"The smallest singular value of random rectangular matrices with no moment assumptions on entries","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Konstantin E. Tikhomirov","submitted_at":"2014-09-29T00:53:01Z","abstract_excerpt":"Let $\\delta>1$ and $\\beta>0$ be some real numbers. We prove that there are positive $u,v,N_0$ depending only on $\\beta$ and $\\delta$ with the following property: for any $N,n$ such that $N\\ge \\max(N_0,\\delta n)$, any $N\\times n$ random matrix $A=(a_{ij})$ with i.i.d. entries satisfying $\\sup\\limits_{\\lambda\\in {\\mathbb R}}{\\mathbb P}\\bigl\\{|a_{11}-\\lambda|\\le 1\\bigr\\}\\le 1-\\beta$ and any non-random $N\\times n$ matrix $B$, the smallest singular value $s_n$ of $A+B$ satisfies ${\\mathbb P}\\bigl\\{s_n(A+B)\\le u\\sqrt{N}\\bigr\\}\\le \\exp(-vN)$. The result holds without any moment assumptions on distrib"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1409.7975","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"}