{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:QHM3FKUC4XFMS7NSNBQJAN3NZR","short_pith_number":"pith:QHM3FKUC","canonical_record":{"source":{"id":"1510.01025","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2015-10-05T04:14:22Z","cross_cats_sorted":["cs.LG","cs.NA","math.NA"],"title_canon_sha256":"ca4c444a28a7bfac35c338cb68ea9337430a8fdafa50daa9bdcb748d9c71498f","abstract_canon_sha256":"ab6080d62849a34c4003caa48cab14474c6061cf9670988afb3f2217c29596f0"},"schema_version":"1.0"},"canonical_sha256":"81d9b2aa82e5cac97db2686090376dcc622f356f6ee3468131ac50ebaf446ea7","source":{"kind":"arxiv","id":"1510.01025","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1510.01025","created_at":"2026-05-18T01:31:04Z"},{"alias_kind":"arxiv_version","alias_value":"1510.01025v1","created_at":"2026-05-18T01:31:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1510.01025","created_at":"2026-05-18T01:31:04Z"},{"alias_kind":"pith_short_12","alias_value":"QHM3FKUC4XFM","created_at":"2026-05-18T12:29:37Z"},{"alias_kind":"pith_short_16","alias_value":"QHM3FKUC4XFMS7NS","created_at":"2026-05-18T12:29:37Z"},{"alias_kind":"pith_short_8","alias_value":"QHM3FKUC","created_at":"2026-05-18T12:29:37Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:QHM3FKUC4XFMS7NSNBQJAN3NZR","target":"record","payload":{"canonical_record":{"source":{"id":"1510.01025","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2015-10-05T04:14:22Z","cross_cats_sorted":["cs.LG","cs.NA","math.NA"],"title_canon_sha256":"ca4c444a28a7bfac35c338cb68ea9337430a8fdafa50daa9bdcb748d9c71498f","abstract_canon_sha256":"ab6080d62849a34c4003caa48cab14474c6061cf9670988afb3f2217c29596f0"},"schema_version":"1.0"},"canonical_sha256":"81d9b2aa82e5cac97db2686090376dcc622f356f6ee3468131ac50ebaf446ea7","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:31:04.964260Z","signature_b64":"/26UkR61Gi1U53mC5oG1HlDrLDfUS91vTo6NF613SzaUxJGmjkdiuFUskiv8L3IW38anmsf61tWi9YKM+YlvCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"81d9b2aa82e5cac97db2686090376dcc622f356f6ee3468131ac50ebaf446ea7","last_reissued_at":"2026-05-18T01:31:04.963741Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:31:04.963741Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1510.01025","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:31:04Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"LbfgUy/STeXOARgFoQRFuz37iSdQVL0zNKHQ7XZqS1suq4c9peLO8PkmgomR6jDufucZanMICCk12KI2keGPAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T22:58:50.580163Z"},"content_sha256":"bb602eb72d2979fbd59e0348fd860dd95697d865797eeb6d0d3e1ad9ebe11c05","schema_version":"1.0","event_id":"sha256:bb602eb72d2979fbd59e0348fd860dd95697d865797eeb6d0d3e1ad9ebe11c05"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:QHM3FKUC4XFMS7NSNBQJAN3NZR","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Quadratic Optimization with Orthogonality Constraints: Explicit Lojasiewicz Exponent and Linear Convergence of Line-Search Methods","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.LG","cs.NA","math.NA"],"primary_cat":"math.OC","authors_text":"Anthony Man-Cho So, Huikang Liu, Weijie Wu","submitted_at":"2015-10-05T04:14:22Z","abstract_excerpt":"A fundamental class of matrix optimization problems that arise in many areas of science and engineering is that of quadratic optimization with orthogonality constraints. Such problems can be solved using line-search methods on the Stiefel manifold, which are known to converge globally under mild conditions. To determine the convergence rate of these methods, we give an explicit estimate of the exponent in a Lojasiewicz inequality for the (non-convex) set of critical points of the aforementioned class of problems. By combining such an estimate with known arguments, we are able to establish the "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.01025","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:31:04Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"szfYoBUf7ajeX/wb8mENDCNmE2ee598BnYltp6nIbrFNw7x5mOVMp1u5w8QWMnMYMcUtdPzOxw+/5WD4ioh1BQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T22:58:50.580555Z"},"content_sha256":"c3be2158724202ced371611804aa765df2a7fe3606d8229a9a3f2a9cf9520d55","schema_version":"1.0","event_id":"sha256:c3be2158724202ced371611804aa765df2a7fe3606d8229a9a3f2a9cf9520d55"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/QHM3FKUC4XFMS7NSNBQJAN3NZR/bundle.json","state_url":"https://pith.science/pith/QHM3FKUC4XFMS7NSNBQJAN3NZR/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/QHM3FKUC4XFMS7NSNBQJAN3NZR/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":"2026-05-27T22:58:50Z","links":{"resolver":"https://pith.science/pith/QHM3FKUC4XFMS7NSNBQJAN3NZR","bundle":"https://pith.science/pith/QHM3FKUC4XFMS7NSNBQJAN3NZR/bundle.json","state":"https://pith.science/pith/QHM3FKUC4XFMS7NSNBQJAN3NZR/state.json","well_known_bundle":"https://pith.science/.well-known/pith/QHM3FKUC4XFMS7NSNBQJAN3NZR/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:QHM3FKUC4XFMS7NSNBQJAN3NZR","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":"ab6080d62849a34c4003caa48cab14474c6061cf9670988afb3f2217c29596f0","cross_cats_sorted":["cs.LG","cs.NA","math.NA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2015-10-05T04:14:22Z","title_canon_sha256":"ca4c444a28a7bfac35c338cb68ea9337430a8fdafa50daa9bdcb748d9c71498f"},"schema_version":"1.0","source":{"id":"1510.01025","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1510.01025","created_at":"2026-05-18T01:31:04Z"},{"alias_kind":"arxiv_version","alias_value":"1510.01025v1","created_at":"2026-05-18T01:31:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1510.01025","created_at":"2026-05-18T01:31:04Z"},{"alias_kind":"pith_short_12","alias_value":"QHM3FKUC4XFM","created_at":"2026-05-18T12:29:37Z"},{"alias_kind":"pith_short_16","alias_value":"QHM3FKUC4XFMS7NS","created_at":"2026-05-18T12:29:37Z"},{"alias_kind":"pith_short_8","alias_value":"QHM3FKUC","created_at":"2026-05-18T12:29:37Z"}],"graph_snapshots":[{"event_id":"sha256:c3be2158724202ced371611804aa765df2a7fe3606d8229a9a3f2a9cf9520d55","target":"graph","created_at":"2026-05-18T01:31:04Z","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"},"paper":{"abstract_excerpt":"A fundamental class of matrix optimization problems that arise in many areas of science and engineering is that of quadratic optimization with orthogonality constraints. Such problems can be solved using line-search methods on the Stiefel manifold, which are known to converge globally under mild conditions. To determine the convergence rate of these methods, we give an explicit estimate of the exponent in a Lojasiewicz inequality for the (non-convex) set of critical points of the aforementioned class of problems. By combining such an estimate with known arguments, we are able to establish the ","authors_text":"Anthony Man-Cho So, Huikang Liu, Weijie Wu","cross_cats":["cs.LG","cs.NA","math.NA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2015-10-05T04:14:22Z","title":"Quadratic Optimization with Orthogonality Constraints: Explicit Lojasiewicz Exponent and Linear Convergence of Line-Search Methods"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.01025","kind":"arxiv","version":1},"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:bb602eb72d2979fbd59e0348fd860dd95697d865797eeb6d0d3e1ad9ebe11c05","target":"record","created_at":"2026-05-18T01:31:04Z","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":"ab6080d62849a34c4003caa48cab14474c6061cf9670988afb3f2217c29596f0","cross_cats_sorted":["cs.LG","cs.NA","math.NA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2015-10-05T04:14:22Z","title_canon_sha256":"ca4c444a28a7bfac35c338cb68ea9337430a8fdafa50daa9bdcb748d9c71498f"},"schema_version":"1.0","source":{"id":"1510.01025","kind":"arxiv","version":1}},"canonical_sha256":"81d9b2aa82e5cac97db2686090376dcc622f356f6ee3468131ac50ebaf446ea7","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"81d9b2aa82e5cac97db2686090376dcc622f356f6ee3468131ac50ebaf446ea7","first_computed_at":"2026-05-18T01:31:04.963741Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:31:04.963741Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"/26UkR61Gi1U53mC5oG1HlDrLDfUS91vTo6NF613SzaUxJGmjkdiuFUskiv8L3IW38anmsf61tWi9YKM+YlvCg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:31:04.964260Z","signed_message":"canonical_sha256_bytes"},"source_id":"1510.01025","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:bb602eb72d2979fbd59e0348fd860dd95697d865797eeb6d0d3e1ad9ebe11c05","sha256:c3be2158724202ced371611804aa765df2a7fe3606d8229a9a3f2a9cf9520d55"],"state_sha256":"6f116f2cad5db05091ded102ffe052cb302faf11c04eac8deacef9cc4cd43026"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3Do2QBPI4K7fLRZgzO4RSCX7AA2Ex5vh6Ht79+1aQH3IlP7uRY0Q+u37o9ExUdDooFxdpYJLqmUKKIFdBsaTBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T22:58:50.583122Z","bundle_sha256":"a5e5f965153138c61d077fcd74010ac21395cd46aa161fcf3b66bbe508c57520"}}