{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2024:DCDWMRH2SPFZXY3DZCZ5XOZLMV","short_pith_number":"pith:DCDWMRH2","schema_version":"1.0","canonical_sha256":"18876644fa93cb9be363c8b3dbbb2b657480701f17ea898fee4d13b660312f6b","source":{"kind":"arxiv","id":"2408.14389","version":2},"attestation_state":"computed","paper":{"title":"The smallest singular value of inhomogenous random rectangular matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.PR","authors_text":"Manuel Fernandez, Max Dabagia","submitted_at":"2024-08-26T16:18:17Z","abstract_excerpt":"Let $A \\in \\mathbb{R}^{N \\times n}$ ($N \\geq n$) be a random matrix with with independent entries that have mean 0 variance 1 and bounded $2+\\beta$ moment. We show that the smallest singular value $\\sigma_n(A)$ satisfies\n  \\[\n  \\Pr \\left(\\sigma_n(A) \\leq \\varepsilon(\\sqrt{N+1} - \\sqrt{n})\\right) \\leq (C\\varepsilon)^{N-n+1} + e^{-cN},\n  \\]\n  for all $\\varepsilon > 0$, where $c,C$ depend only on $\\beta$ and the $2+\\beta$ moment. This extends earlier results of Rudelson and Vershynin, who showed that such lower tail estimates held for rectangular matrices with i.i.d. mean 0 subgaussian entries. W"},"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":"2408.14389","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2024-08-26T16:18:17Z","cross_cats_sorted":["math.MG"],"title_canon_sha256":"15d50d2ae1f25b9b18cb4d3e662eb76a82ac22ccd80dde5a7993c46e5d27113c","abstract_canon_sha256":"0b91b8efc69da82d363fbf71b7fa4096682a228c40011c2d3a8ecc024d415150"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:43:07.142436Z","signature_b64":"bZ+c/rGWIjI07AcfLs8SUWcpEXfeGwbtR69kG0csP8hPMxL6GiuH9FTCnUnyYHOLlFvqU0ZfrdjK0g/lLgXKCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"18876644fa93cb9be363c8b3dbbb2b657480701f17ea898fee4d13b660312f6b","last_reissued_at":"2026-07-05T11:43:07.141954Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:43:07.141954Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The smallest singular value of inhomogenous random rectangular matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.PR","authors_text":"Manuel Fernandez, Max Dabagia","submitted_at":"2024-08-26T16:18:17Z","abstract_excerpt":"Let $A \\in \\mathbb{R}^{N \\times n}$ ($N \\geq n$) be a random matrix with with independent entries that have mean 0 variance 1 and bounded $2+\\beta$ moment. We show that the smallest singular value $\\sigma_n(A)$ satisfies\n  \\[\n  \\Pr \\left(\\sigma_n(A) \\leq \\varepsilon(\\sqrt{N+1} - \\sqrt{n})\\right) \\leq (C\\varepsilon)^{N-n+1} + e^{-cN},\n  \\]\n  for all $\\varepsilon > 0$, where $c,C$ depend only on $\\beta$ and the $2+\\beta$ moment. This extends earlier results of Rudelson and Vershynin, who showed that such lower tail estimates held for rectangular matrices with i.i.d. mean 0 subgaussian entries. W"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2408.14389","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2408.14389/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2408.14389","created_at":"2026-07-05T11:43:07.142011+00:00"},{"alias_kind":"arxiv_version","alias_value":"2408.14389v2","created_at":"2026-07-05T11:43:07.142011+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2408.14389","created_at":"2026-07-05T11:43:07.142011+00:00"},{"alias_kind":"pith_short_12","alias_value":"DCDWMRH2SPFZ","created_at":"2026-07-05T11:43:07.142011+00:00"},{"alias_kind":"pith_short_16","alias_value":"DCDWMRH2SPFZXY3D","created_at":"2026-07-05T11:43:07.142011+00:00"},{"alias_kind":"pith_short_8","alias_value":"DCDWMRH2","created_at":"2026-07-05T11:43:07.142011+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":2,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2607.02169","citing_title":"A note on \"The volume of random simplices from elliptical distributions in high dimension\"","ref_index":4,"is_internal_anchor":false},{"citing_arxiv_id":"2503.08139","citing_title":"The eigenvalue gap of inhomogeneous symmetric discrete random matrix","ref_index":7,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/DCDWMRH2SPFZXY3DZCZ5XOZLMV","json":"https://pith.science/pith/DCDWMRH2SPFZXY3DZCZ5XOZLMV.json","graph_json":"https://pith.science/api/pith-number/DCDWMRH2SPFZXY3DZCZ5XOZLMV/graph.json","events_json":"https://pith.science/api/pith-number/DCDWMRH2SPFZXY3DZCZ5XOZLMV/events.json","paper":"https://pith.science/paper/DCDWMRH2"},"agent_actions":{"view_html":"https://pith.science/pith/DCDWMRH2SPFZXY3DZCZ5XOZLMV","download_json":"https://pith.science/pith/DCDWMRH2SPFZXY3DZCZ5XOZLMV.json","view_paper":"https://pith.science/paper/DCDWMRH2","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2408.14389&json=true","fetch_graph":"https://pith.science/api/pith-number/DCDWMRH2SPFZXY3DZCZ5XOZLMV/graph.json","fetch_events":"https://pith.science/api/pith-number/DCDWMRH2SPFZXY3DZCZ5XOZLMV/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/DCDWMRH2SPFZXY3DZCZ5XOZLMV/action/timestamp_anchor","attest_storage":"https://pith.science/pith/DCDWMRH2SPFZXY3DZCZ5XOZLMV/action/storage_attestation","attest_author":"https://pith.science/pith/DCDWMRH2SPFZXY3DZCZ5XOZLMV/action/author_attestation","sign_citation":"https://pith.science/pith/DCDWMRH2SPFZXY3DZCZ5XOZLMV/action/citation_signature","submit_replication":"https://pith.science/pith/DCDWMRH2SPFZXY3DZCZ5XOZLMV/action/replication_record"}},"created_at":"2026-07-05T11:43:07.142011+00:00","updated_at":"2026-07-05T11:43:07.142011+00:00"}