{"paper":{"title":"The smallest singular value of inhomogenous random rectangular matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.PR","authors_text":"Manuel Fernandez, Max Dabagia","submitted_at":"2024-08-26T16:18:17Z","abstract_excerpt":"Let $A \\in \\mathbb{R}^{N \\times n}$ ($N \\geq n$) be a random matrix with with independent entries that have mean 0 variance 1 and bounded $2+\\beta$ moment. We show that the smallest singular value $\\sigma_n(A)$ satisfies\n  \\[\n  \\Pr \\left(\\sigma_n(A) \\leq \\varepsilon(\\sqrt{N+1} - \\sqrt{n})\\right) \\leq (C\\varepsilon)^{N-n+1} + e^{-cN},\n  \\]\n  for all $\\varepsilon > 0$, where $c,C$ depend only on $\\beta$ and the $2+\\beta$ moment. This extends earlier results of Rudelson and Vershynin, who showed that such lower tail estimates held for rectangular matrices with i.i.d. mean 0 subgaussian entries. W"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2408.14389","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2408.14389/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"}