{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2017:GTHCLPWQWIKBOONNMTDQDZTVDR","short_pith_number":"pith:GTHCLPWQ","schema_version":"1.0","canonical_sha256":"34ce25bed0b2141739ad64c701e6751c456f47a8eb119a0c33fef4cafa8b0c2d","source":{"kind":"arxiv","id":"1702.04050","version":2},"attestation_state":"computed","paper":{"title":"The smallest singular value of deformed random rectangular matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Fan Yang","submitted_at":"2017-02-14T02:36:08Z","abstract_excerpt":"We prove an estimate on the smallest singular value of a multiplicatively and additively deformed random rectangular matrix. Suppose $n\\le N \\le M \\le \\Lambda N$ for some constant $\\Lambda \\ge 1$. Let $X$ be an $M\\times n$ random matrix with independent and identically distributed entries, which have zero mean, unit variance and arbitrarily high moments. Let $T$ be an $N\\times M$ deterministic matrix with comparable singular values $c\\le s_{N}(T) \\le s_{1}(T) \\le c^{-1}$ for some constant $c>0$. Let $A$ be an $N\\times n$ deterministic matrix with $\\|A\\|=O(\\sqrt{N})$. Then we prove that for any"},"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":"1702.04050","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2017-02-14T02:36:08Z","cross_cats_sorted":[],"title_canon_sha256":"7c3a35d0fed3f8a91feac5d2e5d9a4a45280f62493ae0cf4d26da7a22de14af8","abstract_canon_sha256":"232bd97e0c97903516b1618cbf1a64e327e78deb455c6c4cb413576d52c93af7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:03:19.875426Z","signature_b64":"aAQwGqmnpgSMsOWrNNIHr90btlnz/ltePYSVMrPWidVkxUBPiJrFgMFVaagwhEK5ZbqJ7JnkE7qDk0aSDpF5CQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"34ce25bed0b2141739ad64c701e6751c456f47a8eb119a0c33fef4cafa8b0c2d","last_reissued_at":"2026-05-18T00:03:19.874894Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:03:19.874894Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The smallest singular value of deformed random rectangular matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Fan Yang","submitted_at":"2017-02-14T02:36:08Z","abstract_excerpt":"We prove an estimate on the smallest singular value of a multiplicatively and additively deformed random rectangular matrix. Suppose $n\\le N \\le M \\le \\Lambda N$ for some constant $\\Lambda \\ge 1$. Let $X$ be an $M\\times n$ random matrix with independent and identically distributed entries, which have zero mean, unit variance and arbitrarily high moments. Let $T$ be an $N\\times M$ deterministic matrix with comparable singular values $c\\le s_{N}(T) \\le s_{1}(T) \\le c^{-1}$ for some constant $c>0$. Let $A$ be an $N\\times n$ deterministic matrix with $\\|A\\|=O(\\sqrt{N})$. Then we prove that for any"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1702.04050","kind":"arxiv","version":2},"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":"1702.04050","created_at":"2026-05-18T00:03:19.874977+00:00"},{"alias_kind":"arxiv_version","alias_value":"1702.04050v2","created_at":"2026-05-18T00:03:19.874977+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1702.04050","created_at":"2026-05-18T00:03:19.874977+00:00"},{"alias_kind":"pith_short_12","alias_value":"GTHCLPWQWIKB","created_at":"2026-05-18T12:31:18.294218+00:00"},{"alias_kind":"pith_short_16","alias_value":"GTHCLPWQWIKBOONN","created_at":"2026-05-18T12:31:18.294218+00:00"},{"alias_kind":"pith_short_8","alias_value":"GTHCLPWQ","created_at":"2026-05-18T12:31:18.294218+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/GTHCLPWQWIKBOONNMTDQDZTVDR","json":"https://pith.science/pith/GTHCLPWQWIKBOONNMTDQDZTVDR.json","graph_json":"https://pith.science/api/pith-number/GTHCLPWQWIKBOONNMTDQDZTVDR/graph.json","events_json":"https://pith.science/api/pith-number/GTHCLPWQWIKBOONNMTDQDZTVDR/events.json","paper":"https://pith.science/paper/GTHCLPWQ"},"agent_actions":{"view_html":"https://pith.science/pith/GTHCLPWQWIKBOONNMTDQDZTVDR","download_json":"https://pith.science/pith/GTHCLPWQWIKBOONNMTDQDZTVDR.json","view_paper":"https://pith.science/paper/GTHCLPWQ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1702.04050&json=true","fetch_graph":"https://pith.science/api/pith-number/GTHCLPWQWIKBOONNMTDQDZTVDR/graph.json","fetch_events":"https://pith.science/api/pith-number/GTHCLPWQWIKBOONNMTDQDZTVDR/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/GTHCLPWQWIKBOONNMTDQDZTVDR/action/timestamp_anchor","attest_storage":"https://pith.science/pith/GTHCLPWQWIKBOONNMTDQDZTVDR/action/storage_attestation","attest_author":"https://pith.science/pith/GTHCLPWQWIKBOONNMTDQDZTVDR/action/author_attestation","sign_citation":"https://pith.science/pith/GTHCLPWQWIKBOONNMTDQDZTVDR/action/citation_signature","submit_replication":"https://pith.science/pith/GTHCLPWQWIKBOONNMTDQDZTVDR/action/replication_record"}},"created_at":"2026-05-18T00:03:19.874977+00:00","updated_at":"2026-05-18T00:03:19.874977+00:00"}