{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:N3WGOG5WOQEL76WT3URQWWZYDC","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"9968ccb5018567990bf0d1bb9edaca76b3da8294ca1442f98270d2dedd446cf2","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-12-12T18:37:03Z","title_canon_sha256":"90ddc1eca470c571d91dcd6a132b054d28c4c45dd4648c04d9648aa4155a60c3"},"schema_version":"1.0","source":{"id":"1312.3580","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1312.3580","created_at":"2026-05-18T03:04:50Z"},{"alias_kind":"arxiv_version","alias_value":"1312.3580v1","created_at":"2026-05-18T03:04:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1312.3580","created_at":"2026-05-18T03:04:50Z"},{"alias_kind":"pith_short_12","alias_value":"N3WGOG5WOQEL","created_at":"2026-05-18T12:27:52Z"},{"alias_kind":"pith_short_16","alias_value":"N3WGOG5WOQEL76WT","created_at":"2026-05-18T12:27:52Z"},{"alias_kind":"pith_short_8","alias_value":"N3WGOG5W","created_at":"2026-05-18T12:27:52Z"}],"graph_snapshots":[{"event_id":"sha256:2dc8577ddf526d529075f199004e621dd3f6d536c597ea1535699b6bb9172a54","target":"graph","created_at":"2026-05-18T03:04:50Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"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 ","authors_text":"Shahar Mendelson, Vladimir Koltchinskii","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-12-12T18:37:03Z","title":"Bounding the smallest singular value of a random matrix without concentration"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1312.3580","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:f1ce4ffcf2623f8636901d7a50c685113c31475a5d966a96047068b708013015","target":"record","created_at":"2026-05-18T03:04:50Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"9968ccb5018567990bf0d1bb9edaca76b3da8294ca1442f98270d2dedd446cf2","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-12-12T18:37:03Z","title_canon_sha256":"90ddc1eca470c571d91dcd6a132b054d28c4c45dd4648c04d9648aa4155a60c3"},"schema_version":"1.0","source":{"id":"1312.3580","kind":"arxiv","version":1}},"canonical_sha256":"6eec671bb67408bffad3dd230b5b38188bdeda671a2eee2ceec6171b800c22f5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6eec671bb67408bffad3dd230b5b38188bdeda671a2eee2ceec6171b800c22f5","first_computed_at":"2026-05-18T03:04:50.492274Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:04:50.492274Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"uqfG3SJ+TujhEuCQiLF6oMFtb50lJ/OA5jTMUOjopPkrEK1JNHeyZAHBBj1js3RQFW11Y/+WaGgQnouHDwtcCw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:04:50.493179Z","signed_message":"canonical_sha256_bytes"},"source_id":"1312.3580","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f1ce4ffcf2623f8636901d7a50c685113c31475a5d966a96047068b708013015","sha256:2dc8577ddf526d529075f199004e621dd3f6d536c597ea1535699b6bb9172a54"],"state_sha256":"79c7655a00abb07597642dd59bccee22c28e332d0a188c8c3a0f93bb040cb78a"}