{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2020:KYH5PZP4SN4K6MA3QBZEP4ZK3U","short_pith_number":"pith:KYH5PZP4","canonical_record":{"source":{"id":"2005.06976","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2020-05-14T13:48:52Z","cross_cats_sorted":["cs.NA"],"title_canon_sha256":"0af620bcbb5c37ae8317ebef7f72571e85ebec5d3aa651ebf07880b8fe077058","abstract_canon_sha256":"2a90abf20620881bcf2a9b6e34fa4d90d373005332700971b9908fe7e37ea1f0"},"schema_version":"1.0"},"canonical_sha256":"560fd7e5fc9378af301b807247f32add36dbcebf260b1b0679911e5867e9e196","source":{"kind":"arxiv","id":"2005.06976","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2005.06976","created_at":"2026-06-01T01:03:37Z"},{"alias_kind":"arxiv_version","alias_value":"2005.06976v2","created_at":"2026-06-01T01:03:37Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2005.06976","created_at":"2026-06-01T01:03:37Z"},{"alias_kind":"pith_short_12","alias_value":"KYH5PZP4SN4K","created_at":"2026-06-01T01:03:37Z"},{"alias_kind":"pith_short_16","alias_value":"KYH5PZP4SN4K6MA3","created_at":"2026-06-01T01:03:37Z"},{"alias_kind":"pith_short_8","alias_value":"KYH5PZP4","created_at":"2026-06-01T01:03:37Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2020:KYH5PZP4SN4K6MA3QBZEP4ZK3U","target":"record","payload":{"canonical_record":{"source":{"id":"2005.06976","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2020-05-14T13:48:52Z","cross_cats_sorted":["cs.NA"],"title_canon_sha256":"0af620bcbb5c37ae8317ebef7f72571e85ebec5d3aa651ebf07880b8fe077058","abstract_canon_sha256":"2a90abf20620881bcf2a9b6e34fa4d90d373005332700971b9908fe7e37ea1f0"},"schema_version":"1.0"},"canonical_sha256":"560fd7e5fc9378af301b807247f32add36dbcebf260b1b0679911e5867e9e196","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-01T01:03:37.995727Z","signature_b64":"k9TDtRz1CiFUIF43W95x1+/b4ySjdeUBq7KEACVgjxu/i07qUKjcVtOxJsiGiK/ct243jujRbDvcErNygvyYBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"560fd7e5fc9378af301b807247f32add36dbcebf260b1b0679911e5867e9e196","last_reissued_at":"2026-06-01T01:03:37.994954Z","signature_status":"signed_v1","first_computed_at":"2026-06-01T01:03:37.994954Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2005.06976","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-06-01T01:03:37Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"YReVyGKvIMtPO8beyCVhSHD7wCzviHCW3lkxkDISplAAkWDKSESuJBVopkMKQGvS3ue2FJeMjVuXl8A8kSCFCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T08:20:24.087137Z"},"content_sha256":"47d1dafc8bb364ed3c76017f8d1587f17a199c403a13623db2ac9080728a9fc1","schema_version":"1.0","event_id":"sha256:47d1dafc8bb364ed3c76017f8d1587f17a199c403a13623db2ac9080728a9fc1"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2020:KYH5PZP4SN4K6MA3QBZEP4ZK3U","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Riemannian multigrid line search for low-rank problems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.NA"],"primary_cat":"math.NA","authors_text":"Bart Vandereycken, Marco Sutti","submitted_at":"2020-05-14T13:48:52Z","abstract_excerpt":"Large-scale optimization problems arising from the discretization of problems involving PDEs sometimes admit solutions that can be well approximated by low-rank matrices. In this paper, we will exploit this low-rank approximation property by solving the optimization problem directly over the set of low-rank matrices. In particular, we introduce a new multilevel algorithm, where the optimization variable is constrained to the Riemannian manifold of fixed-rank matrices. In contrast to most other multilevel algorithms where the rank is chosen adaptively on each level in order to control the pertu"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2005.06976","kind":"arxiv","version":2},"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/2005.06976/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":"2026-06-01T01:03:37Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"b3FUEoq2gEhiu4bTBSVhO1OjD54R26MaZffY69yIvRZ1+GJsdYu7ZJ4qo2pL7ppZN6/4AgpfOdPs8KEOzvN8Dw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T08:20:24.087529Z"},"content_sha256":"50f11a693a630d21fb2e39f31f85e289c82fdbb03b54e46c639a95575f5bf5e8","schema_version":"1.0","event_id":"sha256:50f11a693a630d21fb2e39f31f85e289c82fdbb03b54e46c639a95575f5bf5e8"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/KYH5PZP4SN4K6MA3QBZEP4ZK3U/bundle.json","state_url":"https://pith.science/pith/KYH5PZP4SN4K6MA3QBZEP4ZK3U/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/KYH5PZP4SN4K6MA3QBZEP4ZK3U/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":"2026-06-26T08:20:24Z","links":{"resolver":"https://pith.science/pith/KYH5PZP4SN4K6MA3QBZEP4ZK3U","bundle":"https://pith.science/pith/KYH5PZP4SN4K6MA3QBZEP4ZK3U/bundle.json","state":"https://pith.science/pith/KYH5PZP4SN4K6MA3QBZEP4ZK3U/state.json","well_known_bundle":"https://pith.science/.well-known/pith/KYH5PZP4SN4K6MA3QBZEP4ZK3U/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2020:KYH5PZP4SN4K6MA3QBZEP4ZK3U","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":"2a90abf20620881bcf2a9b6e34fa4d90d373005332700971b9908fe7e37ea1f0","cross_cats_sorted":["cs.NA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2020-05-14T13:48:52Z","title_canon_sha256":"0af620bcbb5c37ae8317ebef7f72571e85ebec5d3aa651ebf07880b8fe077058"},"schema_version":"1.0","source":{"id":"2005.06976","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2005.06976","created_at":"2026-06-01T01:03:37Z"},{"alias_kind":"arxiv_version","alias_value":"2005.06976v2","created_at":"2026-06-01T01:03:37Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2005.06976","created_at":"2026-06-01T01:03:37Z"},{"alias_kind":"pith_short_12","alias_value":"KYH5PZP4SN4K","created_at":"2026-06-01T01:03:37Z"},{"alias_kind":"pith_short_16","alias_value":"KYH5PZP4SN4K6MA3","created_at":"2026-06-01T01:03:37Z"},{"alias_kind":"pith_short_8","alias_value":"KYH5PZP4","created_at":"2026-06-01T01:03:37Z"}],"graph_snapshots":[{"event_id":"sha256:50f11a693a630d21fb2e39f31f85e289c82fdbb03b54e46c639a95575f5bf5e8","target":"graph","created_at":"2026-06-01T01:03:37Z","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/2005.06976/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Large-scale optimization problems arising from the discretization of problems involving PDEs sometimes admit solutions that can be well approximated by low-rank matrices. In this paper, we will exploit this low-rank approximation property by solving the optimization problem directly over the set of low-rank matrices. In particular, we introduce a new multilevel algorithm, where the optimization variable is constrained to the Riemannian manifold of fixed-rank matrices. In contrast to most other multilevel algorithms where the rank is chosen adaptively on each level in order to control the pertu","authors_text":"Bart Vandereycken, Marco Sutti","cross_cats":["cs.NA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2020-05-14T13:48:52Z","title":"Riemannian multigrid line search for low-rank problems"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2005.06976","kind":"arxiv","version":2},"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":"sha256:47d1dafc8bb364ed3c76017f8d1587f17a199c403a13623db2ac9080728a9fc1","target":"record","created_at":"2026-06-01T01:03:37Z","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":"2a90abf20620881bcf2a9b6e34fa4d90d373005332700971b9908fe7e37ea1f0","cross_cats_sorted":["cs.NA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2020-05-14T13:48:52Z","title_canon_sha256":"0af620bcbb5c37ae8317ebef7f72571e85ebec5d3aa651ebf07880b8fe077058"},"schema_version":"1.0","source":{"id":"2005.06976","kind":"arxiv","version":2}},"canonical_sha256":"560fd7e5fc9378af301b807247f32add36dbcebf260b1b0679911e5867e9e196","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"560fd7e5fc9378af301b807247f32add36dbcebf260b1b0679911e5867e9e196","first_computed_at":"2026-06-01T01:03:37.994954Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-01T01:03:37.994954Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"k9TDtRz1CiFUIF43W95x1+/b4ySjdeUBq7KEACVgjxu/i07qUKjcVtOxJsiGiK/ct243jujRbDvcErNygvyYBg==","signature_status":"signed_v1","signed_at":"2026-06-01T01:03:37.995727Z","signed_message":"canonical_sha256_bytes"},"source_id":"2005.06976","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:47d1dafc8bb364ed3c76017f8d1587f17a199c403a13623db2ac9080728a9fc1","sha256:50f11a693a630d21fb2e39f31f85e289c82fdbb03b54e46c639a95575f5bf5e8"],"state_sha256":"0f032c42cb3cdeade771e07535fd973ef021e00e8876de98168083a373b7b018"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"HKbCFStQ7l6hBwZo1CTzqRU5sI0mJFkkFuUhAkV+vOyO8AAcT9AV8PNlQhNX9B6vZQacz2emTb2OA2dy0aB/Cg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-26T08:20:24.090064Z","bundle_sha256":"882c126578266afb8bf2f30b67bbd0481bcbd92275bfd4f5384e230327bcbf1d"}}