{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:MP2A75XT73GWLPYLKBCOLGM2RX","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":"7b1b068218e8642d86223b363abaaaa397e742363c00c6d1048861093d5b447b","cross_cats_sorted":["cs.LG","math.NA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2014-02-21T12:49:51Z","title_canon_sha256":"5959ddd024a379dc7e3a5aae6ac573d65eda82b326eb07749fceb24596c80576"},"schema_version":"1.0","source":{"id":"1402.5284","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1402.5284","created_at":"2026-05-18T02:18:13Z"},{"alias_kind":"arxiv_version","alias_value":"1402.5284v3","created_at":"2026-05-18T02:18:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1402.5284","created_at":"2026-05-18T02:18:13Z"},{"alias_kind":"pith_short_12","alias_value":"MP2A75XT73GW","created_at":"2026-05-18T12:28:38Z"},{"alias_kind":"pith_short_16","alias_value":"MP2A75XT73GWLPYL","created_at":"2026-05-18T12:28:38Z"},{"alias_kind":"pith_short_8","alias_value":"MP2A75XT","created_at":"2026-05-18T12:28:38Z"}],"graph_snapshots":[{"event_id":"sha256:06be7da3b271391b81a1f317d37278f8ea5055ce7de56a52e255f9083bfadbc9","target":"graph","created_at":"2026-05-18T02:18:13Z","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":"The aim of this paper is to derive convergence results for projected line-search methods on the real-algebraic variety $\\mathcal{M}_{\\le k}$ of real $m \\times n$ matrices of rank at most $k$. Such methods extend Riemannian optimization methods, which are successfully used on the smooth manifold $\\mathcal{M}_k$ of rank-$k$ matrices, to its closure by taking steps along gradient-related directions in the tangent cone, and afterwards projecting back to $\\mathcal{M}_{\\le k}$. Considering such a method circumvents the difficulties which arise from the nonclosedness and the unbounded curvature of $\\","authors_text":"Andr\\'e Uschmajew, Reinhold Schneider","cross_cats":["cs.LG","math.NA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2014-02-21T12:49:51Z","title":"Convergence results for projected line-search methods on varieties of low-rank matrices via \\L{}ojasiewicz inequality"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1402.5284","kind":"arxiv","version":3},"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":"sha256:3ace55771e90a6a78f009d7b66f71391adf958b1945caead30e79d7b4c432fa1","target":"record","created_at":"2026-05-18T02:18:13Z","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":"7b1b068218e8642d86223b363abaaaa397e742363c00c6d1048861093d5b447b","cross_cats_sorted":["cs.LG","math.NA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2014-02-21T12:49:51Z","title_canon_sha256":"5959ddd024a379dc7e3a5aae6ac573d65eda82b326eb07749fceb24596c80576"},"schema_version":"1.0","source":{"id":"1402.5284","kind":"arxiv","version":3}},"canonical_sha256":"63f40ff6f3fecd65bf0b5044e5999a8de5b35c8fad9576ca6949eea524820c85","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"63f40ff6f3fecd65bf0b5044e5999a8de5b35c8fad9576ca6949eea524820c85","first_computed_at":"2026-05-18T02:18:13.721348Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:18:13.721348Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"b4EdZctOMEA9T6Kw3VwvxYvuMwKO5s96HHwsJ3mRsRGfwRGohpMSSz7sZ9z1qTQFGjjZgoOoujGe3C1dBMiHCg==","signature_status":"signed_v1","signed_at":"2026-05-18T02:18:13.721812Z","signed_message":"canonical_sha256_bytes"},"source_id":"1402.5284","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3ace55771e90a6a78f009d7b66f71391adf958b1945caead30e79d7b4c432fa1","sha256:06be7da3b271391b81a1f317d37278f8ea5055ce7de56a52e255f9083bfadbc9"],"state_sha256":"ec7f8bea6005a222e82cc76c61252ade898423b7b97204df637bd964cf5ea39c"}