{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2020:KV3ODEP3SGSS7AASPCD5XR6LTJ","short_pith_number":"pith:KV3ODEP3","schema_version":"1.0","canonical_sha256":"5576e191fb91a52f80127887dbc7cb9a4b226a80174d9e0c22c9a75de6bdb7db","source":{"kind":"arxiv","id":"2009.10071","version":4},"attestation_state":"computed","paper":{"title":"QR and LQ Decomposition Matrix Backpropagation Algorithms for Square, Wide, and Deep -- Real or Complex -- Matrices and Their Software Implementation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.LG","cs.MS","cs.NA","stat.ML"],"primary_cat":"math.NA","authors_text":"Denisa A.O. Roberts, Lucas R. Roberts","submitted_at":"2020-09-19T21:03:37Z","abstract_excerpt":"This article presents matrix backpropagation algorithms for the QR decomposition of matrices $A_{m, n}$, that are either square (m = n), wide (m < n), or deep (m > n), with rank $k = min(m, n)$. Furthermore, we derive novel matrix backpropagation results for the pivoted (full-rank) QR decomposition and for the LQ decomposition of deep input matrices. Differentiable QR decomposition offers a numerically stable, computationally efficient method to solve least squares problems frequently encountered in machine learning and computer vision. Other use cases such as graph learning and network compre"},"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":"2009.10071","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2020-09-19T21:03:37Z","cross_cats_sorted":["cs.LG","cs.MS","cs.NA","stat.ML"],"title_canon_sha256":"ab3aeaf573df225745ed8207e1c4e44362fae430933e543f6f1b4a7c2b7f6d91","abstract_canon_sha256":"1ba15c0631776925bb8de55eb93de4cf6ab4f26e0c272860edcc079de32a90a4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T01:58:42.558281Z","signature_b64":"D6L8h28/kSFhu7T8DHFRhAY0/FIjlo4MpG2egHxBpbNFy7+aXqYo5Sq2y6RhSEFtc9DdrpKpFpuhJ3EtI+HxBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5576e191fb91a52f80127887dbc7cb9a4b226a80174d9e0c22c9a75de6bdb7db","last_reissued_at":"2026-07-05T01:58:42.557859Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T01:58:42.557859Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"QR and LQ Decomposition Matrix Backpropagation Algorithms for Square, Wide, and Deep -- Real or Complex -- Matrices and Their Software Implementation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.LG","cs.MS","cs.NA","stat.ML"],"primary_cat":"math.NA","authors_text":"Denisa A.O. Roberts, Lucas R. Roberts","submitted_at":"2020-09-19T21:03:37Z","abstract_excerpt":"This article presents matrix backpropagation algorithms for the QR decomposition of matrices $A_{m, n}$, that are either square (m = n), wide (m < n), or deep (m > n), with rank $k = min(m, n)$. Furthermore, we derive novel matrix backpropagation results for the pivoted (full-rank) QR decomposition and for the LQ decomposition of deep input matrices. Differentiable QR decomposition offers a numerically stable, computationally efficient method to solve least squares problems frequently encountered in machine learning and computer vision. Other use cases such as graph learning and network compre"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2009.10071","kind":"arxiv","version":4},"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/2009.10071/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":"2009.10071","created_at":"2026-07-05T01:58:42.557917+00:00"},{"alias_kind":"arxiv_version","alias_value":"2009.10071v4","created_at":"2026-07-05T01:58:42.557917+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2009.10071","created_at":"2026-07-05T01:58:42.557917+00:00"},{"alias_kind":"pith_short_12","alias_value":"KV3ODEP3SGSS","created_at":"2026-07-05T01:58:42.557917+00:00"},{"alias_kind":"pith_short_16","alias_value":"KV3ODEP3SGSS7AAS","created_at":"2026-07-05T01:58:42.557917+00:00"},{"alias_kind":"pith_short_8","alias_value":"KV3ODEP3","created_at":"2026-07-05T01:58:42.557917+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2507.08570","citing_title":"Photonic processor benchmarking for variational quantum process tomography","ref_index":27,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/KV3ODEP3SGSS7AASPCD5XR6LTJ","json":"https://pith.science/pith/KV3ODEP3SGSS7AASPCD5XR6LTJ.json","graph_json":"https://pith.science/api/pith-number/KV3ODEP3SGSS7AASPCD5XR6LTJ/graph.json","events_json":"https://pith.science/api/pith-number/KV3ODEP3SGSS7AASPCD5XR6LTJ/events.json","paper":"https://pith.science/paper/KV3ODEP3"},"agent_actions":{"view_html":"https://pith.science/pith/KV3ODEP3SGSS7AASPCD5XR6LTJ","download_json":"https://pith.science/pith/KV3ODEP3SGSS7AASPCD5XR6LTJ.json","view_paper":"https://pith.science/paper/KV3ODEP3","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2009.10071&json=true","fetch_graph":"https://pith.science/api/pith-number/KV3ODEP3SGSS7AASPCD5XR6LTJ/graph.json","fetch_events":"https://pith.science/api/pith-number/KV3ODEP3SGSS7AASPCD5XR6LTJ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/KV3ODEP3SGSS7AASPCD5XR6LTJ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/KV3ODEP3SGSS7AASPCD5XR6LTJ/action/storage_attestation","attest_author":"https://pith.science/pith/KV3ODEP3SGSS7AASPCD5XR6LTJ/action/author_attestation","sign_citation":"https://pith.science/pith/KV3ODEP3SGSS7AASPCD5XR6LTJ/action/citation_signature","submit_replication":"https://pith.science/pith/KV3ODEP3SGSS7AASPCD5XR6LTJ/action/replication_record"}},"created_at":"2026-07-05T01:58:42.557917+00:00","updated_at":"2026-07-05T01:58:42.557917+00:00"}