{"paper":{"title":"A Riemannian gradient descent method for optimization on the indefinite Stiefel manifold","license":"http://creativecommons.org/licenses/by/4.0/","headline":"A Riemannian gradient descent method on the indefinite Stiefel manifold X^T A X = J converges globally to critical points.","cross_cats":[],"primary_cat":"math.OC","authors_text":"Dinh Van Tiep, Nguyen Thanh Son","submitted_at":"2024-10-29T14:27:52Z","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"},"claims":{"count":4,"items":[{"kind":"strongest_claim","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.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","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.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","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.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"A Riemannian gradient descent method on the indefinite Stiefel manifold X^T A X = J converges globally to critical points.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"f7064b0ca2d481f71350a3b3a87bb90862ec466815650a963e6189297dcf67bf"},"source":{"id":"2410.22068","kind":"arxiv","version":3},"verdict":{"id":"7da68cc3-9e3e-4ba9-9717-908963b731f0","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-23T18:42:39.149989Z","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.","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","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.","pith_extraction_headline":"A Riemannian gradient descent method on the indefinite Stiefel manifold X^T A X = J converges globally to critical points."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2410.22068/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":2,"snapshot_sha256":"37a4794c44e933de7ee105fbe89a61f3e2598c4965025c03733436edc1a0d9ab"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}