{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:JAJKLDURWP3MP6I4PRGEFV5TGQ","short_pith_number":"pith:JAJKLDUR","schema_version":"1.0","canonical_sha256":"4812a58e91b3f6c7f91c7c4c42d7b3343d5a7556fac1db80d093a91e0b196ece","source":{"kind":"arxiv","id":"1507.03525","version":3},"attestation_state":"computed","paper":{"title":"Invertibility of Sparse non-Hermitian matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Anirban Basak, Mark Rudelson","submitted_at":"2015-07-13T17:15:32Z","abstract_excerpt":"We consider a class of sparse random matrices of the form $A_n =(\\xi_{i,j}\\delta_{i,j})_{i,j=1}^n$, where $\\{\\xi_{i,j}\\}$ are i.i.d.~centered random variables, and $\\{\\delta_{i,j}\\}$ are i.i.d.~Bernoulli random variables taking value $1$ with probability $p_n$, and prove a quantitative estimate on the smallest singular value for $p_n = \\Omega(\\frac{\\log n}{n})$, under a suitable assumption on the spectral norm of the matrices. This establishes the invertibility of a large class of sparse matrices. For $p_n =\\Omega( n^{-\\alpha})$ with some $\\alpha \\in (0,1)$, we deduce that the condition number"},"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":"1507.03525","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-07-13T17:15:32Z","cross_cats_sorted":[],"title_canon_sha256":"ea40e26e04fe3bf9955e3ade8532a5b326d20d90833ba2072a48072785489f4a","abstract_canon_sha256":"30dd1146303a2a15bc51f8c0cca7d7d33fbaf99df295622df23548d35ce0e717"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:51:30.627684Z","signature_b64":"xJZC4dEpubHyzursInwShcVg5W9qxU1oqu6D4UTz5+amsJAFiDu8w+yexs035uOPSTQzxOPtr7RboNAVw30/Bw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4812a58e91b3f6c7f91c7c4c42d7b3343d5a7556fac1db80d093a91e0b196ece","last_reissued_at":"2026-05-18T00:51:30.627065Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:51:30.627065Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Invertibility of Sparse non-Hermitian matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Anirban Basak, Mark Rudelson","submitted_at":"2015-07-13T17:15:32Z","abstract_excerpt":"We consider a class of sparse random matrices of the form $A_n =(\\xi_{i,j}\\delta_{i,j})_{i,j=1}^n$, where $\\{\\xi_{i,j}\\}$ are i.i.d.~centered random variables, and $\\{\\delta_{i,j}\\}$ are i.i.d.~Bernoulli random variables taking value $1$ with probability $p_n$, and prove a quantitative estimate on the smallest singular value for $p_n = \\Omega(\\frac{\\log n}{n})$, under a suitable assumption on the spectral norm of the matrices. This establishes the invertibility of a large class of sparse matrices. For $p_n =\\Omega( n^{-\\alpha})$ with some $\\alpha \\in (0,1)$, we deduce that the condition number"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.03525","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":"1507.03525","created_at":"2026-05-18T00:51:30.627154+00:00"},{"alias_kind":"arxiv_version","alias_value":"1507.03525v3","created_at":"2026-05-18T00:51:30.627154+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1507.03525","created_at":"2026-05-18T00:51:30.627154+00:00"},{"alias_kind":"pith_short_12","alias_value":"JAJKLDURWP3M","created_at":"2026-05-18T12:29:27.538025+00:00"},{"alias_kind":"pith_short_16","alias_value":"JAJKLDURWP3MP6I4","created_at":"2026-05-18T12:29:27.538025+00:00"},{"alias_kind":"pith_short_8","alias_value":"JAJKLDUR","created_at":"2026-05-18T12:29:27.538025+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/JAJKLDURWP3MP6I4PRGEFV5TGQ","json":"https://pith.science/pith/JAJKLDURWP3MP6I4PRGEFV5TGQ.json","graph_json":"https://pith.science/api/pith-number/JAJKLDURWP3MP6I4PRGEFV5TGQ/graph.json","events_json":"https://pith.science/api/pith-number/JAJKLDURWP3MP6I4PRGEFV5TGQ/events.json","paper":"https://pith.science/paper/JAJKLDUR"},"agent_actions":{"view_html":"https://pith.science/pith/JAJKLDURWP3MP6I4PRGEFV5TGQ","download_json":"https://pith.science/pith/JAJKLDURWP3MP6I4PRGEFV5TGQ.json","view_paper":"https://pith.science/paper/JAJKLDUR","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1507.03525&json=true","fetch_graph":"https://pith.science/api/pith-number/JAJKLDURWP3MP6I4PRGEFV5TGQ/graph.json","fetch_events":"https://pith.science/api/pith-number/JAJKLDURWP3MP6I4PRGEFV5TGQ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/JAJKLDURWP3MP6I4PRGEFV5TGQ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/JAJKLDURWP3MP6I4PRGEFV5TGQ/action/storage_attestation","attest_author":"https://pith.science/pith/JAJKLDURWP3MP6I4PRGEFV5TGQ/action/author_attestation","sign_citation":"https://pith.science/pith/JAJKLDURWP3MP6I4PRGEFV5TGQ/action/citation_signature","submit_replication":"https://pith.science/pith/JAJKLDURWP3MP6I4PRGEFV5TGQ/action/replication_record"}},"created_at":"2026-05-18T00:51:30.627154+00:00","updated_at":"2026-05-18T00:51:30.627154+00:00"}