{"paper":{"title":"Bounding the smallest singular value of a random matrix without concentration","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Shahar Mendelson, Vladimir Koltchinskii","submitted_at":"2013-12-12T18:37:03Z","abstract_excerpt":"Given $X$ a random vector in ${\\mathbb{R}}^n$, set $X_1,...,X_N$ to be independent copies of $X$ and let $\\Gamma=\\frac{1}{\\sqrt{N}}\\sum_{i=1}^N <X_i,\\cdot>e_i$ be the matrix whose rows are $\\frac{X_1}{\\sqrt{N}},\\dots, \\frac{X_N}{\\sqrt{N}}$.\n  We obtain sharp probabilistic lower bounds on the smallest singular value $\\lambda_{\\min}(\\Gamma)$ in a rather general situation, and in particular, under the assumption that $X$ is an isotropic random vector for which $\\sup_{t\\in S^{n-1}}{\\mathbb{E}}|<t,X>|^{2+\\eta} \\leq L$ for some $L,\\eta>0$. Our results imply that a Bai-Yin type lower bound holds for "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1312.3580","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"}