{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2025:MNMWOQZQEH5HO4WFIR4VILJKRL","short_pith_number":"pith:MNMWOQZQ","schema_version":"1.0","canonical_sha256":"635967433021fa7772c54479542d2a8aece1023d5f1a00c06bb46672fc647493","source":{"kind":"arxiv","id":"2509.04386","version":1},"attestation_state":"computed","paper":{"title":"Randomized biorthogonalization through a two-sided Gram-Schmidt process","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.NA"],"primary_cat":"math.NA","authors_text":"Igor Simunec, Laura Grigori, Lorenzo Piccinini","submitted_at":"2025-09-04T16:49:50Z","abstract_excerpt":"We propose and analyze a randomized two-sided Gram-Schmidt process for the biorthogonalization of two given matrices $X, Y \\in\\mathbb{R}^{n\\times m}$. The algorithm aims to find two matrices $Q, P \\in\\mathbb{R}^{n\\times m}$ such that ${\\rm range}(X) = {\\rm range}(Q)$, ${\\rm range}(Y) = {\\rm range}(P)$ and $(\\Omega Q)^T \\Omega P = I$, where $\\Omega \\in\\mathbb{R}^{s \\times n}$ is a sketching matrix satisfying an oblivious subspace $\\varepsilon$-embedding property; in other words, the biorthogonality condition on the columns of $Q$ and $P$ is replaced by an equivalent condition on their sketches."},"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":"2509.04386","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2025-09-04T16:49:50Z","cross_cats_sorted":["cs.NA"],"title_canon_sha256":"1bde946fa4be6a0f729bce5bec5881f5409fc7675fbf0389ed0503843135219d","abstract_canon_sha256":"9fdf7055bcc64f14aeaff83b8f5f063b813e4c2e86060c25f96467bb86317477"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T12:04:58.707009Z","signature_b64":"H5kiYwpYVzAtDHjpOrMl66LzC0zPey4zYv4v8oh5IB3KnsnXXsC2PmugTiguWPFSKJQzf/m6iqIFiPorRfocAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"635967433021fa7772c54479542d2a8aece1023d5f1a00c06bb46672fc647493","last_reissued_at":"2026-07-05T12:04:58.706464Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T12:04:58.706464Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Randomized biorthogonalization through a two-sided Gram-Schmidt process","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.NA"],"primary_cat":"math.NA","authors_text":"Igor Simunec, Laura Grigori, Lorenzo Piccinini","submitted_at":"2025-09-04T16:49:50Z","abstract_excerpt":"We propose and analyze a randomized two-sided Gram-Schmidt process for the biorthogonalization of two given matrices $X, Y \\in\\mathbb{R}^{n\\times m}$. The algorithm aims to find two matrices $Q, P \\in\\mathbb{R}^{n\\times m}$ such that ${\\rm range}(X) = {\\rm range}(Q)$, ${\\rm range}(Y) = {\\rm range}(P)$ and $(\\Omega Q)^T \\Omega P = I$, where $\\Omega \\in\\mathbb{R}^{s \\times n}$ is a sketching matrix satisfying an oblivious subspace $\\varepsilon$-embedding property; in other words, the biorthogonality condition on the columns of $Q$ and $P$ is replaced by an equivalent condition on their sketches."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2509.04386","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2509.04386/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":"2509.04386","created_at":"2026-07-05T12:04:58.706533+00:00"},{"alias_kind":"arxiv_version","alias_value":"2509.04386v1","created_at":"2026-07-05T12:04:58.706533+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2509.04386","created_at":"2026-07-05T12:04:58.706533+00:00"},{"alias_kind":"pith_short_12","alias_value":"MNMWOQZQEH5H","created_at":"2026-07-05T12:04:58.706533+00:00"},{"alias_kind":"pith_short_16","alias_value":"MNMWOQZQEH5HO4WF","created_at":"2026-07-05T12:04:58.706533+00:00"},{"alias_kind":"pith_short_8","alias_value":"MNMWOQZQ","created_at":"2026-07-05T12:04:58.706533+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/MNMWOQZQEH5HO4WFIR4VILJKRL","json":"https://pith.science/pith/MNMWOQZQEH5HO4WFIR4VILJKRL.json","graph_json":"https://pith.science/api/pith-number/MNMWOQZQEH5HO4WFIR4VILJKRL/graph.json","events_json":"https://pith.science/api/pith-number/MNMWOQZQEH5HO4WFIR4VILJKRL/events.json","paper":"https://pith.science/paper/MNMWOQZQ"},"agent_actions":{"view_html":"https://pith.science/pith/MNMWOQZQEH5HO4WFIR4VILJKRL","download_json":"https://pith.science/pith/MNMWOQZQEH5HO4WFIR4VILJKRL.json","view_paper":"https://pith.science/paper/MNMWOQZQ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2509.04386&json=true","fetch_graph":"https://pith.science/api/pith-number/MNMWOQZQEH5HO4WFIR4VILJKRL/graph.json","fetch_events":"https://pith.science/api/pith-number/MNMWOQZQEH5HO4WFIR4VILJKRL/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/MNMWOQZQEH5HO4WFIR4VILJKRL/action/timestamp_anchor","attest_storage":"https://pith.science/pith/MNMWOQZQEH5HO4WFIR4VILJKRL/action/storage_attestation","attest_author":"https://pith.science/pith/MNMWOQZQEH5HO4WFIR4VILJKRL/action/author_attestation","sign_citation":"https://pith.science/pith/MNMWOQZQEH5HO4WFIR4VILJKRL/action/citation_signature","submit_replication":"https://pith.science/pith/MNMWOQZQEH5HO4WFIR4VILJKRL/action/replication_record"}},"created_at":"2026-07-05T12:04:58.706533+00:00","updated_at":"2026-07-05T12:04:58.706533+00:00"}