{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:EQLQ7OAGSN2XRQULXTPKZUOAIT","short_pith_number":"pith:EQLQ7OAG","schema_version":"1.0","canonical_sha256":"24170fb806937578c28bbcdeacd1c044c4cd32a90a7e0447eb02a625c1ccdec5","source":{"kind":"arxiv","id":"1601.06259","version":1},"attestation_state":"computed","paper":{"title":"Minimax Lower Bounds for Linear Independence Testing","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.IT","cs.LG","math.IT","math.ST","stat.TH"],"primary_cat":"stat.ML","authors_text":"Aaditya Ramdas, Aarti Singh, David Isenberg, Larry Wasserman","submitted_at":"2016-01-23T10:20:58Z","abstract_excerpt":"Linear independence testing is a fundamental information-theoretic and statistical problem that can be posed as follows: given $n$ points $\\{(X_i,Y_i)\\}^n_{i=1}$ from a $p+q$ dimensional multivariate distribution where $X_i \\in \\mathbb{R}^p$ and $Y_i \\in\\mathbb{R}^q$, determine whether $a^T X$ and $b^T Y$ are uncorrelated for every $a \\in \\mathbb{R}^p, b\\in \\mathbb{R}^q$ or not. We give minimax lower bound for this problem (when $p+q,n \\to \\infty$, $(p+q)/n \\leq \\kappa < \\infty$, without sparsity assumptions). In summary, our results imply that $n$ must be at least as large as $\\sqrt {pq}/\\|\\S"},"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":"1601.06259","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"stat.ML","submitted_at":"2016-01-23T10:20:58Z","cross_cats_sorted":["cs.IT","cs.LG","math.IT","math.ST","stat.TH"],"title_canon_sha256":"e0e3a0bcb36c96012b64b6ef54b6e7275a71c186de3381ac46e09160fa351f21","abstract_canon_sha256":"008e77a42e9308aaa0ddebeb39ac661c57fae79f1a1acb6db4116bb6b2649fca"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:22:04.923795Z","signature_b64":"7fBwuPRqmuSrMFZO4o3OOH5ML1ampJcSixVS3usujhPCLNVakVZbI6VMiHgp32qlmgAkpVMn9y/SPC8MjLmeCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"24170fb806937578c28bbcdeacd1c044c4cd32a90a7e0447eb02a625c1ccdec5","last_reissued_at":"2026-05-18T01:22:04.923044Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:22:04.923044Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Minimax Lower Bounds for Linear Independence Testing","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.IT","cs.LG","math.IT","math.ST","stat.TH"],"primary_cat":"stat.ML","authors_text":"Aaditya Ramdas, Aarti Singh, David Isenberg, Larry Wasserman","submitted_at":"2016-01-23T10:20:58Z","abstract_excerpt":"Linear independence testing is a fundamental information-theoretic and statistical problem that can be posed as follows: given $n$ points $\\{(X_i,Y_i)\\}^n_{i=1}$ from a $p+q$ dimensional multivariate distribution where $X_i \\in \\mathbb{R}^p$ and $Y_i \\in\\mathbb{R}^q$, determine whether $a^T X$ and $b^T Y$ are uncorrelated for every $a \\in \\mathbb{R}^p, b\\in \\mathbb{R}^q$ or not. We give minimax lower bound for this problem (when $p+q,n \\to \\infty$, $(p+q)/n \\leq \\kappa < \\infty$, without sparsity assumptions). In summary, our results imply that $n$ must be at least as large as $\\sqrt {pq}/\\|\\S"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.06259","kind":"arxiv","version":1},"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":"1601.06259","created_at":"2026-05-18T01:22:04.923174+00:00"},{"alias_kind":"arxiv_version","alias_value":"1601.06259v1","created_at":"2026-05-18T01:22:04.923174+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1601.06259","created_at":"2026-05-18T01:22:04.923174+00:00"},{"alias_kind":"pith_short_12","alias_value":"EQLQ7OAGSN2X","created_at":"2026-05-18T12:30:12.583610+00:00"},{"alias_kind":"pith_short_16","alias_value":"EQLQ7OAGSN2XRQUL","created_at":"2026-05-18T12:30:12.583610+00:00"},{"alias_kind":"pith_short_8","alias_value":"EQLQ7OAG","created_at":"2026-05-18T12:30:12.583610+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/EQLQ7OAGSN2XRQULXTPKZUOAIT","json":"https://pith.science/pith/EQLQ7OAGSN2XRQULXTPKZUOAIT.json","graph_json":"https://pith.science/api/pith-number/EQLQ7OAGSN2XRQULXTPKZUOAIT/graph.json","events_json":"https://pith.science/api/pith-number/EQLQ7OAGSN2XRQULXTPKZUOAIT/events.json","paper":"https://pith.science/paper/EQLQ7OAG"},"agent_actions":{"view_html":"https://pith.science/pith/EQLQ7OAGSN2XRQULXTPKZUOAIT","download_json":"https://pith.science/pith/EQLQ7OAGSN2XRQULXTPKZUOAIT.json","view_paper":"https://pith.science/paper/EQLQ7OAG","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1601.06259&json=true","fetch_graph":"https://pith.science/api/pith-number/EQLQ7OAGSN2XRQULXTPKZUOAIT/graph.json","fetch_events":"https://pith.science/api/pith-number/EQLQ7OAGSN2XRQULXTPKZUOAIT/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/EQLQ7OAGSN2XRQULXTPKZUOAIT/action/timestamp_anchor","attest_storage":"https://pith.science/pith/EQLQ7OAGSN2XRQULXTPKZUOAIT/action/storage_attestation","attest_author":"https://pith.science/pith/EQLQ7OAGSN2XRQULXTPKZUOAIT/action/author_attestation","sign_citation":"https://pith.science/pith/EQLQ7OAGSN2XRQULXTPKZUOAIT/action/citation_signature","submit_replication":"https://pith.science/pith/EQLQ7OAGSN2XRQULXTPKZUOAIT/action/replication_record"}},"created_at":"2026-05-18T01:22:04.923174+00:00","updated_at":"2026-05-18T01:22:04.923174+00:00"}