{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:HKF5WRBM53GPA7QX2KFCNYK355","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":"48d2ce38b6bdcd7238a36e27dfe1c61e8840d44f9b44b0bf67278dbf90f6ad98","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.OC","submitted_at":"2024-10-29T14:27:52Z","title_canon_sha256":"f69e1e2cc76c4e7715e2a6efefaf7633599c9dce86c4866e6de0609cdaea1a57"},"schema_version":"1.0","source":{"id":"2410.22068","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2410.22068","created_at":"2026-05-26T01:03:09Z"},{"alias_kind":"arxiv_version","alias_value":"2410.22068v3","created_at":"2026-05-26T01:03:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2410.22068","created_at":"2026-05-26T01:03:09Z"},{"alias_kind":"pith_short_12","alias_value":"HKF5WRBM53GP","created_at":"2026-05-26T01:03:09Z"},{"alias_kind":"pith_short_16","alias_value":"HKF5WRBM53GPA7QX","created_at":"2026-05-26T01:03:09Z"},{"alias_kind":"pith_short_8","alias_value":"HKF5WRBM","created_at":"2026-05-26T01:03:09Z"}],"graph_snapshots":[{"event_id":"sha256:8ca1c00db03bb98d8bc03cb813973cc9a6283fe076d5743ed5e2dd1b97fa29b4","target":"graph","created_at":"2026-05-26T01:03:09Z","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":4,"items":[{"attestation":"unclaimed","claim_id":"C1","kind":"strongest_claim","source":"verdict.strongest_claim","status":"machine_extracted","text":"We ... suggest a Riemannian gradient descent method using the attained materials, whose global convergence is guaranteed. Our results not only cover the known cases, the orthogonal and generalized Stiefel manifolds, but also provide a Riemannian optimization solution for other constrained problems which has not been investigated."},{"attestation":"unclaimed","claim_id":"C2","kind":"weakest_assumption","source":"verdict.weakest_assumption","status":"machine_extracted","text":"The feasible set X^T A X = J constitutes a differentiable manifold (the indefinite Stiefel manifold) that admits a Riemannian metric allowing construction of the associated geometric structure and a well-defined Cayley retraction."},{"attestation":"unclaimed","claim_id":"C3","kind":"one_line_summary","source":"verdict.one_line_summary","status":"machine_extracted","text":"Develops Riemannian gradient descent with Cayley retraction on the indefinite Stiefel manifold X^T A X = J, proves global convergence, generalizes orthogonal cases, and applies to eigenvalue problems and Procrustes-type matrix equations."},{"attestation":"unclaimed","claim_id":"C4","kind":"headline","source":"verdict.pith_extraction.headline","status":"machine_extracted","text":"A Riemannian gradient descent method on the indefinite Stiefel manifold X^T A X = J converges globally to critical points."}],"snapshot_sha256":"f7064b0ca2d481f71350a3b3a87bb90862ec466815650a963e6189297dcf67bf"},"formal_canon":{"evidence_count":2,"snapshot_sha256":"37a4794c44e933de7ee105fbe89a61f3e2598c4965025c03733436edc1a0d9ab"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2410.22068/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We consider the optimization problem with a generally quadratic matrix constraint of the form $X^TAX = J$, where $A$ is a given nonsingular, symmetric $n\\times n$ matrix and $J$ is a given $k\\times k$ symmetric matrix, with $k\\leq n$, satisfying $J^2 = I_k$. Since the feasible set constitutes a differentiable manifold, called the indefinite Stiefel manifold, we approach this problem within the framework of Riemannian optimization. Namely, we first equip the manifold with a Riemannian metric and construct the associated geometric structure, then propose a retraction based on the Cayley transfor","authors_text":"Dinh Van Tiep, Nguyen Thanh Son","cross_cats":[],"headline":"A Riemannian gradient descent method on the indefinite Stiefel manifold X^T A X = J converges globally to critical points.","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.OC","submitted_at":"2024-10-29T14:27:52Z","title":"A Riemannian gradient descent method for optimization on the indefinite Stiefel manifold"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2410.22068","kind":"arxiv","version":3},"verdict":{"created_at":"2026-05-23T18:42:39.149989Z","id":"7da68cc3-9e3e-4ba9-9717-908963b731f0","model_set":{"reader":"grok-4.3"},"one_line_summary":"Develops Riemannian gradient descent with Cayley retraction on the indefinite Stiefel manifold X^T A X = J, proves global convergence, generalizes orthogonal cases, and applies to eigenvalue problems and Procrustes-type matrix equations.","pipeline_version":"pith-pipeline@v0.9.0","pith_extraction_headline":"A Riemannian gradient descent method on the indefinite Stiefel manifold X^T A X = J converges globally to critical points.","strongest_claim":"We ... suggest a Riemannian gradient descent method using the attained materials, whose global convergence is guaranteed. Our results not only cover the known cases, the orthogonal and generalized Stiefel manifolds, but also provide a Riemannian optimization solution for other constrained problems which has not been investigated.","weakest_assumption":"The feasible set X^T A X = J constitutes a differentiable manifold (the indefinite Stiefel manifold) that admits a Riemannian metric allowing construction of the associated geometric structure and a well-defined Cayley retraction."}},"verdict_id":"7da68cc3-9e3e-4ba9-9717-908963b731f0"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:045278887e000a235146be0926537abe54a59912523342a9dd034cb8010deec7","target":"record","created_at":"2026-05-26T01:03:09Z","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":"48d2ce38b6bdcd7238a36e27dfe1c61e8840d44f9b44b0bf67278dbf90f6ad98","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.OC","submitted_at":"2024-10-29T14:27:52Z","title_canon_sha256":"f69e1e2cc76c4e7715e2a6efefaf7633599c9dce86c4866e6de0609cdaea1a57"},"schema_version":"1.0","source":{"id":"2410.22068","kind":"arxiv","version":3}},"canonical_sha256":"3a8bdb442ceeccf07e17d28a26e15bef4662b5875dece7b12a0444faabede767","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3a8bdb442ceeccf07e17d28a26e15bef4662b5875dece7b12a0444faabede767","first_computed_at":"2026-05-26T01:03:09.536728Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-26T01:03:09.536728Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"aLFPQf+2RomjeCCvdSB6whrDCh0EcjeLrQrAYK4S5WomONAMu+aK2CpWOtjLLXsfaPGkKZf5r0JDpzqZAY4wCA==","signature_status":"signed_v1","signed_at":"2026-05-26T01:03:09.537520Z","signed_message":"canonical_sha256_bytes"},"source_id":"2410.22068","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:045278887e000a235146be0926537abe54a59912523342a9dd034cb8010deec7","sha256:8ca1c00db03bb98d8bc03cb813973cc9a6283fe076d5743ed5e2dd1b97fa29b4"],"state_sha256":"85a9fc307baa2f81110b4ee3ac8fe7be35df02595d4c8da66d7a07d54b006783"}