{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2021:A34KPJZGZJ3UB24R7A6NPNRDXV","short_pith_number":"pith:A34KPJZG","canonical_record":{"source":{"id":"2111.14385","kind":"arxiv","version":4},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NA","submitted_at":"2021-11-29T08:56:37Z","cross_cats_sorted":["cs.NA"],"title_canon_sha256":"2b9ce5de5be4a31c6e8c4f1f8f9c1c90d606b009d3dc4dd31ef380635b8e942d","abstract_canon_sha256":"f5a3e219679893a2b87e84ee0351482b955b93379f532e763e4d1fda77c3cece"},"schema_version":"1.0"},"canonical_sha256":"06f8a7a726ca7740eb91f83cd7b623bd7b84730c256c93bcf44b3af838c5cfd5","source":{"kind":"arxiv","id":"2111.14385","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2111.14385","created_at":"2026-07-05T04:40:08Z"},{"alias_kind":"arxiv_version","alias_value":"2111.14385v4","created_at":"2026-07-05T04:40:08Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2111.14385","created_at":"2026-07-05T04:40:08Z"},{"alias_kind":"pith_short_12","alias_value":"A34KPJZGZJ3U","created_at":"2026-07-05T04:40:08Z"},{"alias_kind":"pith_short_16","alias_value":"A34KPJZGZJ3UB24R","created_at":"2026-07-05T04:40:08Z"},{"alias_kind":"pith_short_8","alias_value":"A34KPJZG","created_at":"2026-07-05T04:40:08Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2021:A34KPJZGZJ3UB24R7A6NPNRDXV","target":"record","payload":{"canonical_record":{"source":{"id":"2111.14385","kind":"arxiv","version":4},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NA","submitted_at":"2021-11-29T08:56:37Z","cross_cats_sorted":["cs.NA"],"title_canon_sha256":"2b9ce5de5be4a31c6e8c4f1f8f9c1c90d606b009d3dc4dd31ef380635b8e942d","abstract_canon_sha256":"f5a3e219679893a2b87e84ee0351482b955b93379f532e763e4d1fda77c3cece"},"schema_version":"1.0"},"canonical_sha256":"06f8a7a726ca7740eb91f83cd7b623bd7b84730c256c93bcf44b3af838c5cfd5","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T04:40:08.169024Z","signature_b64":"LaVtvhg3EcSUxps8M1KqjhmqLdAMCIQkJlxdYzp4tEZlDfwQ0LbqR5mnAho1WluPuQbfB6ax2GN/RHH4jVx3DQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"06f8a7a726ca7740eb91f83cd7b623bd7b84730c256c93bcf44b3af838c5cfd5","last_reissued_at":"2026-07-05T04:40:08.168666Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T04:40:08.168666Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2111.14385","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T04:40:08Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"JHF3SL2MrwS+BeLQhaLTwkz2KK2o+vMMvEIV/pqz3ySC/a7Ea8FrFmq/i/t6Ez/7x07SHeg6uNrLnEJNCCkCBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-21T19:59:44.398706Z"},"content_sha256":"e00f233d884760127480b50f7b5272985f27b8231b790a1bac9189bee83c2a0d","schema_version":"1.0","event_id":"sha256:e00f233d884760127480b50f7b5272985f27b8231b790a1bac9189bee83c2a0d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2021:A34KPJZGZJ3UB24R7A6NPNRDXV","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A theory of meta-factorization","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.NA"],"primary_cat":"math.NA","authors_text":"Micha{\\l} P. Karpowicz","submitted_at":"2021-11-29T08:56:37Z","abstract_excerpt":"We introduce meta-factorization, a theory that describes matrix decompositions as solutions of linear matrix equations: the projector and the reconstruction equation. Meta-factorization reconstructs known factorizations, reveals their internal structures, and allows for introducing modifications, as illustrated with SVD, QR, and UTV factorizations. The prospect of meta-factorization also provides insights into computational aspects of generalized matrix inverses and randomized linear algebra algorithms. The relations between the Moore-Penrose pseudoinverse, generalized Nystr\\\"{o}m method, and "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2111.14385","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/2111.14385/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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T04:40:08Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"rKFxtiPxX7LCgGQzifHcSgF8YeVhOY+3NiJUB8kVxy68ZA/Fwc7gayIU/DuBeaW9YEIxwvnjuqdmNpLiDncjDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-21T19:59:44.399219Z"},"content_sha256":"b2b1f8fcd289f11c673d50b0fda75de81e34545f150ac7242779d310663f3229","schema_version":"1.0","event_id":"sha256:b2b1f8fcd289f11c673d50b0fda75de81e34545f150ac7242779d310663f3229"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/A34KPJZGZJ3UB24R7A6NPNRDXV/bundle.json","state_url":"https://pith.science/pith/A34KPJZGZJ3UB24R7A6NPNRDXV/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/A34KPJZGZJ3UB24R7A6NPNRDXV/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-08-21T19:59:44Z","links":{"resolver":"https://pith.science/pith/A34KPJZGZJ3UB24R7A6NPNRDXV","bundle":"https://pith.science/pith/A34KPJZGZJ3UB24R7A6NPNRDXV/bundle.json","state":"https://pith.science/pith/A34KPJZGZJ3UB24R7A6NPNRDXV/state.json","well_known_bundle":"https://pith.science/.well-known/pith/A34KPJZGZJ3UB24R7A6NPNRDXV/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2021:A34KPJZGZJ3UB24R7A6NPNRDXV","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"f5a3e219679893a2b87e84ee0351482b955b93379f532e763e4d1fda77c3cece","cross_cats_sorted":["cs.NA"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NA","submitted_at":"2021-11-29T08:56:37Z","title_canon_sha256":"2b9ce5de5be4a31c6e8c4f1f8f9c1c90d606b009d3dc4dd31ef380635b8e942d"},"schema_version":"1.0","source":{"id":"2111.14385","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2111.14385","created_at":"2026-07-05T04:40:08Z"},{"alias_kind":"arxiv_version","alias_value":"2111.14385v4","created_at":"2026-07-05T04:40:08Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2111.14385","created_at":"2026-07-05T04:40:08Z"},{"alias_kind":"pith_short_12","alias_value":"A34KPJZGZJ3U","created_at":"2026-07-05T04:40:08Z"},{"alias_kind":"pith_short_16","alias_value":"A34KPJZGZJ3UB24R","created_at":"2026-07-05T04:40:08Z"},{"alias_kind":"pith_short_8","alias_value":"A34KPJZG","created_at":"2026-07-05T04:40:08Z"}],"graph_snapshots":[{"event_id":"sha256:b2b1f8fcd289f11c673d50b0fda75de81e34545f150ac7242779d310663f3229","target":"graph","created_at":"2026-07-05T04:40:08Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2111.14385/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We introduce meta-factorization, a theory that describes matrix decompositions as solutions of linear matrix equations: the projector and the reconstruction equation. Meta-factorization reconstructs known factorizations, reveals their internal structures, and allows for introducing modifications, as illustrated with SVD, QR, and UTV factorizations. The prospect of meta-factorization also provides insights into computational aspects of generalized matrix inverses and randomized linear algebra algorithms. The relations between the Moore-Penrose pseudoinverse, generalized Nystr\\\"{o}m method, and ","authors_text":"Micha{\\l} P. Karpowicz","cross_cats":["cs.NA"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NA","submitted_at":"2021-11-29T08:56:37Z","title":"A theory of meta-factorization"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2111.14385","kind":"arxiv","version":4},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:e00f233d884760127480b50f7b5272985f27b8231b790a1bac9189bee83c2a0d","target":"record","created_at":"2026-07-05T04:40:08Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"f5a3e219679893a2b87e84ee0351482b955b93379f532e763e4d1fda77c3cece","cross_cats_sorted":["cs.NA"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NA","submitted_at":"2021-11-29T08:56:37Z","title_canon_sha256":"2b9ce5de5be4a31c6e8c4f1f8f9c1c90d606b009d3dc4dd31ef380635b8e942d"},"schema_version":"1.0","source":{"id":"2111.14385","kind":"arxiv","version":4}},"canonical_sha256":"06f8a7a726ca7740eb91f83cd7b623bd7b84730c256c93bcf44b3af838c5cfd5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"06f8a7a726ca7740eb91f83cd7b623bd7b84730c256c93bcf44b3af838c5cfd5","first_computed_at":"2026-07-05T04:40:08.168666Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T04:40:08.168666Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"LaVtvhg3EcSUxps8M1KqjhmqLdAMCIQkJlxdYzp4tEZlDfwQ0LbqR5mnAho1WluPuQbfB6ax2GN/RHH4jVx3DQ==","signature_status":"signed_v1","signed_at":"2026-07-05T04:40:08.169024Z","signed_message":"canonical_sha256_bytes"},"source_id":"2111.14385","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e00f233d884760127480b50f7b5272985f27b8231b790a1bac9189bee83c2a0d","sha256:b2b1f8fcd289f11c673d50b0fda75de81e34545f150ac7242779d310663f3229"],"state_sha256":"adff1aa63d49c5dc1bd18e2ce6e8ea804a8c0da498197c4c275bac9efbee0eb5"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"hHgiqqCTAM2sROxX3aCQIqO9MI/TlxqUFhUNR5mjTUKTPHu/Z4/ihEWP8SIrpT0QyZDZLN9y5VWBi8CqOcpkAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-21T19:59:44.404498Z","bundle_sha256":"36270b9beb762a1be3082626abc3e9e7fe2cadfe80eb5b5d6d3f556ce2b98380"}}