{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2008:5VOVCUR4LB4DIJ3TRS5VJNAYKB","short_pith_number":"pith:5VOVCUR4","schema_version":"1.0","canonical_sha256":"ed5d51523c58783427738cbb54b4185045208ca60a8faecc2a1cf7f7d4eeea31","source":{"kind":"arxiv","id":"0802.3956","version":4},"attestation_state":"computed","paper":{"title":"The smallest singular value of a random rectangular matrix","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.PR","authors_text":"Mark Rudelson, Roman Vershynin","submitted_at":"2008-02-27T06:36:03Z","abstract_excerpt":"We prove an optimal estimate on the smallest singular value of a random subgaussian matrix, valid for all fixed dimensions. For an N by n matrix A with independent and identically distributed subgaussian entries, the smallest singular value of A is at least of the order \\sqrt{N} - \\sqrt{n-1} with high probability. A sharp estimate on the probability is also obtained."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"0802.3956","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2008-02-27T06:36:03Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"d4ac922dde16dded39d759515a8b6360e38ac116adcaac0ccdcd55f59f7b129c","abstract_canon_sha256":"0f40f1efa9a8b9dfa13f5738feae62079cf2088c2d19799a670b396553753551"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:54:14.248620Z","signature_b64":"xO4przqMdJnyYDR1rVAe6o/qWG2wQzmfAClBPgAKwH0ok2Rix6UlMkc1RcXnRuLEOjgUUKN0Ma/p5DmmwACBDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ed5d51523c58783427738cbb54b4185045208ca60a8faecc2a1cf7f7d4eeea31","last_reissued_at":"2026-05-18T00:54:14.248227Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:54:14.248227Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The smallest singular value of a random rectangular matrix","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.PR","authors_text":"Mark Rudelson, Roman Vershynin","submitted_at":"2008-02-27T06:36:03Z","abstract_excerpt":"We prove an optimal estimate on the smallest singular value of a random subgaussian matrix, valid for all fixed dimensions. For an N by n matrix A with independent and identically distributed subgaussian entries, the smallest singular value of A is at least of the order \\sqrt{N} - \\sqrt{n-1} with high probability. A sharp estimate on the probability is also obtained."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0802.3956","kind":"arxiv","version":4},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"0802.3956","created_at":"2026-05-18T00:54:14.248283+00:00"},{"alias_kind":"arxiv_version","alias_value":"0802.3956v4","created_at":"2026-05-18T00:54:14.248283+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0802.3956","created_at":"2026-05-18T00:54:14.248283+00:00"},{"alias_kind":"pith_short_12","alias_value":"5VOVCUR4LB4D","created_at":"2026-05-18T12:25:56.245647+00:00"},{"alias_kind":"pith_short_16","alias_value":"5VOVCUR4LB4DIJ3T","created_at":"2026-05-18T12:25:56.245647+00:00"},{"alias_kind":"pith_short_8","alias_value":"5VOVCUR4","created_at":"2026-05-18T12:25:56.245647+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2603.26554","citing_title":"Sharp Capacity Scaling of Spectral Optimizers in Learning Associative Memory","ref_index":45,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/5VOVCUR4LB4DIJ3TRS5VJNAYKB","json":"https://pith.science/pith/5VOVCUR4LB4DIJ3TRS5VJNAYKB.json","graph_json":"https://pith.science/api/pith-number/5VOVCUR4LB4DIJ3TRS5VJNAYKB/graph.json","events_json":"https://pith.science/api/pith-number/5VOVCUR4LB4DIJ3TRS5VJNAYKB/events.json","paper":"https://pith.science/paper/5VOVCUR4"},"agent_actions":{"view_html":"https://pith.science/pith/5VOVCUR4LB4DIJ3TRS5VJNAYKB","download_json":"https://pith.science/pith/5VOVCUR4LB4DIJ3TRS5VJNAYKB.json","view_paper":"https://pith.science/paper/5VOVCUR4","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=0802.3956&json=true","fetch_graph":"https://pith.science/api/pith-number/5VOVCUR4LB4DIJ3TRS5VJNAYKB/graph.json","fetch_events":"https://pith.science/api/pith-number/5VOVCUR4LB4DIJ3TRS5VJNAYKB/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/5VOVCUR4LB4DIJ3TRS5VJNAYKB/action/timestamp_anchor","attest_storage":"https://pith.science/pith/5VOVCUR4LB4DIJ3TRS5VJNAYKB/action/storage_attestation","attest_author":"https://pith.science/pith/5VOVCUR4LB4DIJ3TRS5VJNAYKB/action/author_attestation","sign_citation":"https://pith.science/pith/5VOVCUR4LB4DIJ3TRS5VJNAYKB/action/citation_signature","submit_replication":"https://pith.science/pith/5VOVCUR4LB4DIJ3TRS5VJNAYKB/action/replication_record"}},"created_at":"2026-05-18T00:54:14.248283+00:00","updated_at":"2026-05-18T00:54:14.248283+00:00"}