{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2008:7NEOMFLOJ752ZXEFZNXQXQPAOW","short_pith_number":"pith:7NEOMFLO","schema_version":"1.0","canonical_sha256":"fb48e6156e4ffbacdc85cb6f0bc1e0758a4560972301f10f9b1d98928cb293f6","source":{"kind":"arxiv","id":"0805.3167","version":3},"attestation_state":"computed","paper":{"title":"Smooth analysis of the condition number and the least singular value","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Terence Tao, Van Vu","submitted_at":"2008-05-20T21:03:15Z","abstract_excerpt":"Let $\\a$ be a complex random variable with mean zero and bounded variance. Let $N_{n}$ be the random matrix of size $n$ whose entries are iid copies of $\\a$ and $M$ be a fixed matrix of the same size. The goal of this paper is to give a general estimate for the condition number and least singular value of the matrix $M + N_{n}$, generalizing an earlier result of Spielman and Teng for the case when $\\a$ is gaussian.\n  Our investigation reveals an interesting fact that the \"core\" matrix $M$ does play a role on tail bounds for the least singular value of $M+N_{n} $. This does not occur in Spielma"},"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":"0805.3167","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2008-05-20T21:03:15Z","cross_cats_sorted":[],"title_canon_sha256":"6f413c50cb97e0d4a12003d09cb795f59ebe8d2c345a1b041c2863382745a4c8","abstract_canon_sha256":"d9a2b319cec40525c76d24f546ce4289d819ddf003751da5399655678cc91d5f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:44:10.560367Z","signature_b64":"VmjrmaQD7XY38gk3GL3SgSCljnmzytPpwdSMk7TxpqPrGK1uOYg5jRE8kbOOryYAvLvJDVLzsJWW6ZpFwLN7Dg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fb48e6156e4ffbacdc85cb6f0bc1e0758a4560972301f10f9b1d98928cb293f6","last_reissued_at":"2026-05-18T00:44:10.559878Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:44:10.559878Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Smooth analysis of the condition number and the least singular value","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Terence Tao, Van Vu","submitted_at":"2008-05-20T21:03:15Z","abstract_excerpt":"Let $\\a$ be a complex random variable with mean zero and bounded variance. Let $N_{n}$ be the random matrix of size $n$ whose entries are iid copies of $\\a$ and $M$ be a fixed matrix of the same size. The goal of this paper is to give a general estimate for the condition number and least singular value of the matrix $M + N_{n}$, generalizing an earlier result of Spielman and Teng for the case when $\\a$ is gaussian.\n  Our investigation reveals an interesting fact that the \"core\" matrix $M$ does play a role on tail bounds for the least singular value of $M+N_{n} $. This does not occur in Spielma"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0805.3167","kind":"arxiv","version":3},"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":"0805.3167","created_at":"2026-05-18T00:44:10.559942+00:00"},{"alias_kind":"arxiv_version","alias_value":"0805.3167v3","created_at":"2026-05-18T00:44:10.559942+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0805.3167","created_at":"2026-05-18T00:44:10.559942+00:00"},{"alias_kind":"pith_short_12","alias_value":"7NEOMFLOJ752","created_at":"2026-05-18T12:25:56.245647+00:00"},{"alias_kind":"pith_short_16","alias_value":"7NEOMFLOJ752ZXEF","created_at":"2026-05-18T12:25:56.245647+00:00"},{"alias_kind":"pith_short_8","alias_value":"7NEOMFLO","created_at":"2026-05-18T12:25:56.245647+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/7NEOMFLOJ752ZXEFZNXQXQPAOW","json":"https://pith.science/pith/7NEOMFLOJ752ZXEFZNXQXQPAOW.json","graph_json":"https://pith.science/api/pith-number/7NEOMFLOJ752ZXEFZNXQXQPAOW/graph.json","events_json":"https://pith.science/api/pith-number/7NEOMFLOJ752ZXEFZNXQXQPAOW/events.json","paper":"https://pith.science/paper/7NEOMFLO"},"agent_actions":{"view_html":"https://pith.science/pith/7NEOMFLOJ752ZXEFZNXQXQPAOW","download_json":"https://pith.science/pith/7NEOMFLOJ752ZXEFZNXQXQPAOW.json","view_paper":"https://pith.science/paper/7NEOMFLO","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=0805.3167&json=true","fetch_graph":"https://pith.science/api/pith-number/7NEOMFLOJ752ZXEFZNXQXQPAOW/graph.json","fetch_events":"https://pith.science/api/pith-number/7NEOMFLOJ752ZXEFZNXQXQPAOW/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/7NEOMFLOJ752ZXEFZNXQXQPAOW/action/timestamp_anchor","attest_storage":"https://pith.science/pith/7NEOMFLOJ752ZXEFZNXQXQPAOW/action/storage_attestation","attest_author":"https://pith.science/pith/7NEOMFLOJ752ZXEFZNXQXQPAOW/action/author_attestation","sign_citation":"https://pith.science/pith/7NEOMFLOJ752ZXEFZNXQXQPAOW/action/citation_signature","submit_replication":"https://pith.science/pith/7NEOMFLOJ752ZXEFZNXQXQPAOW/action/replication_record"}},"created_at":"2026-05-18T00:44:10.559942+00:00","updated_at":"2026-05-18T00:44:10.559942+00:00"}