{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:KJTR67M5KRKOPNDVEZS7AVDWWI","short_pith_number":"pith:KJTR67M5","schema_version":"1.0","canonical_sha256":"52671f7d9d5454e7b4752665f05476b21cb69053df7504df3b95378cbf2b35ae","source":{"kind":"arxiv","id":"1506.02764","version":1},"attestation_state":"computed","paper":{"title":"Perturbation of linear forms of singular vectors under Gaussian noise","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Dong Xia, Vladimir Koltchinskii","submitted_at":"2015-06-09T03:26:27Z","abstract_excerpt":"Let $A\\in\\mathbb{R}^{m\\times n}$ be a matrix of rank $r$ with singular value decomposition (SVD) $A=\\sum_{k=1}^r\\sigma_k (u_k\\otimes v_k),$ where $\\{\\sigma_k, k=1,\\ldots,r\\}$ are singular values of $A$ (arranged in a non-increasing order) and $u_k\\in {\\mathbb R}^m, v_k\\in {\\mathbb R}^n, k=1,\\ldots, r$ are the corresponding left and right orthonormal singular vectors. Let $\\tilde{A}=A+X$ be a noisy observation of $A,$ where $X\\in\\mathbb{R}^{m\\times n}$ is a random matrix with i.i.d. Gaussian entries, $X_{ij}\\sim\\mathcal{N}(0,\\tau^2),$ and consider its SVD $\\tilde{A}=\\sum_{k=1}^{m\\wedge n}\\tilde"},"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":"1506.02764","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-06-09T03:26:27Z","cross_cats_sorted":[],"title_canon_sha256":"4cac0bb0f97c60f9beb6ed61f9cf726804559b3e38a24b9df13c2871f0304dfd","abstract_canon_sha256":"05e7ceb5f3787b662cd55274c8cf09b46f95c5d7d0804cfdd7a5ee3d0754fe74"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:55:41.020658Z","signature_b64":"Lk0nVOkXv6lp7+Im8+ZBW9LbVKHROvjL8hVecdbLyBwmWqof4wW8XNfCXV3kXZ7IFEqVO0CBxX7f6h+dJuViCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"52671f7d9d5454e7b4752665f05476b21cb69053df7504df3b95378cbf2b35ae","last_reissued_at":"2026-05-18T01:55:41.019939Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:55:41.019939Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Perturbation of linear forms of singular vectors under Gaussian noise","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Dong Xia, Vladimir Koltchinskii","submitted_at":"2015-06-09T03:26:27Z","abstract_excerpt":"Let $A\\in\\mathbb{R}^{m\\times n}$ be a matrix of rank $r$ with singular value decomposition (SVD) $A=\\sum_{k=1}^r\\sigma_k (u_k\\otimes v_k),$ where $\\{\\sigma_k, k=1,\\ldots,r\\}$ are singular values of $A$ (arranged in a non-increasing order) and $u_k\\in {\\mathbb R}^m, v_k\\in {\\mathbb R}^n, k=1,\\ldots, r$ are the corresponding left and right orthonormal singular vectors. Let $\\tilde{A}=A+X$ be a noisy observation of $A,$ where $X\\in\\mathbb{R}^{m\\times n}$ is a random matrix with i.i.d. Gaussian entries, $X_{ij}\\sim\\mathcal{N}(0,\\tau^2),$ and consider its SVD $\\tilde{A}=\\sum_{k=1}^{m\\wedge n}\\tilde"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1506.02764","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":"1506.02764","created_at":"2026-05-18T01:55:41.020055+00:00"},{"alias_kind":"arxiv_version","alias_value":"1506.02764v1","created_at":"2026-05-18T01:55:41.020055+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1506.02764","created_at":"2026-05-18T01:55:41.020055+00:00"},{"alias_kind":"pith_short_12","alias_value":"KJTR67M5KRKO","created_at":"2026-05-18T12:29:29.992203+00:00"},{"alias_kind":"pith_short_16","alias_value":"KJTR67M5KRKOPNDV","created_at":"2026-05-18T12:29:29.992203+00:00"},{"alias_kind":"pith_short_8","alias_value":"KJTR67M5","created_at":"2026-05-18T12:29:29.992203+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/KJTR67M5KRKOPNDVEZS7AVDWWI","json":"https://pith.science/pith/KJTR67M5KRKOPNDVEZS7AVDWWI.json","graph_json":"https://pith.science/api/pith-number/KJTR67M5KRKOPNDVEZS7AVDWWI/graph.json","events_json":"https://pith.science/api/pith-number/KJTR67M5KRKOPNDVEZS7AVDWWI/events.json","paper":"https://pith.science/paper/KJTR67M5"},"agent_actions":{"view_html":"https://pith.science/pith/KJTR67M5KRKOPNDVEZS7AVDWWI","download_json":"https://pith.science/pith/KJTR67M5KRKOPNDVEZS7AVDWWI.json","view_paper":"https://pith.science/paper/KJTR67M5","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1506.02764&json=true","fetch_graph":"https://pith.science/api/pith-number/KJTR67M5KRKOPNDVEZS7AVDWWI/graph.json","fetch_events":"https://pith.science/api/pith-number/KJTR67M5KRKOPNDVEZS7AVDWWI/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/KJTR67M5KRKOPNDVEZS7AVDWWI/action/timestamp_anchor","attest_storage":"https://pith.science/pith/KJTR67M5KRKOPNDVEZS7AVDWWI/action/storage_attestation","attest_author":"https://pith.science/pith/KJTR67M5KRKOPNDVEZS7AVDWWI/action/author_attestation","sign_citation":"https://pith.science/pith/KJTR67M5KRKOPNDVEZS7AVDWWI/action/citation_signature","submit_replication":"https://pith.science/pith/KJTR67M5KRKOPNDVEZS7AVDWWI/action/replication_record"}},"created_at":"2026-05-18T01:55:41.020055+00:00","updated_at":"2026-05-18T01:55:41.020055+00:00"}