{"paper":{"title":"Approximate Spielman-Teng theorems for the least singular value of random combinatorial matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.PR","authors_text":"Vishesh Jain","submitted_at":"2019-04-24T01:20:24Z","abstract_excerpt":"An approximate Spielman-Teng theorem for the least singular value $s_n(M_n)$ of a random $n\\times n$ square matrix $M_n$ is a statement of the following form: there exist constants $C,c >0$ such that for all $\\eta \\geq 0$, $\\Pr(s_n(M_n) \\leq \\eta) \\lesssim n^{C}\\eta + \\exp(-n^{c})$. The goal of this paper is to develop a simple and novel framework for proving such results for discrete random matrices. As an application, we prove an approximate Spielman-Teng theorem for $\\{0,1\\}$-valued matrices, each of whose rows is an independent vector with exactly $n/2$ zero components. This improves on pr"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1904.10592","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"}