{"as_of":"2026-08-16T16:53:00Z","caps":{"database_statements":6,"inbound":100,"outbound":100},"context_digest":"sha256:bad865d672d886e7cfdda3cb014cfefac35bca73ab68dc305e5012719140d292","coverage":[{"denominator":22,"lane":"reference_resolution","note":"Typed states for the displayed outbound observations.","records_observed":22,"source":"paper_references, paper_reference_links","source_observed_at":"2026-08-11T12:47:23.996654Z","state":"measured"},{"denominator":22,"lane":"standing_notices","note":"One-hop event checks from named stored sources.","records_observed":22,"source":"scholarly_work_events, retraction_status_cache","source_observed_at":"2026-08-16T06:30:59.297886+00:00","state":"measured"},{"denominator":0,"lane":"inbound_itemization","note":"Pith citing papers itemized under the disclosed page cap.","records_observed":0,"source":"paper_references, paper_reference_links","source_observed_at":null,"state":"measured"},{"denominator":1,"lane":"external_citation_measurements","note":"A source-named dated measurement, never combined with another source.","records_observed":0,"source":"cited_works","source_observed_at":null,"state":"measured"}],"external_citation_measurements":[],"inbound":[],"links":{"evidence":"/evidence","html":"/paper/2412.13990/citation-record","integrity":"/paper/2412.13990/integrity","json":"/paper/2412.13990/citation-record.json","paper":"/paper/2412.13990"},"outbound":[{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-11T12:47:24.185569Z","title":"Optimization algo- rithms on matrix manifolds","venue":null,"work_id":"a49d6a55-d56e-4e57-b70f-e7b9afb5ecb8","year":2009},"citing_paper":{"arxiv_id":"2412.13990","last_updated":"2025-01-19T18:43:04Z","snapshot_observed_at":"2026-08-14T21:30:28.592183Z","submitted_at":"2024-12-18T16:10:44Z","title":"A geodesic convexity-like structure for the polar decomposition of a square matrix","version":2},"reference_index":1,"source":"pdf_text","source_observed_at":"2026-08-11T12:47:23.937372Z"},"links":{"citing_paper":"/paper/2412.13990"},"observation_digest":"sha256:ec577468d9247b741dc43f0d247853b28883162db18260cc9b8d6dea6754f058","observation_id":"b1a45141-9944-432d-b788-ddc6fee31196","resolution":{"observed_at":"2026-08-11T12:47:24.188298Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-16T06:30:59.297886+00:00","source":"crossref"},{"observed_at":"2026-08-16T06:30:54.164669+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-11T12:47:24.178169Z","title":"Generalized procrustes analysis and its ap plications in pho- togrammetry","venue":null,"work_id":"57b92b0e-628d-4947-af9f-72f41a00ed63","year":2003},"citing_paper":{"arxiv_id":"2412.13990","last_updated":"2025-01-19T18:43:04Z","snapshot_observed_at":"2026-08-14T21:30:28.592183Z","submitted_at":"2024-12-18T16:10:44Z","title":"A geodesic convexity-like structure for the polar decomposition of a square matrix","version":2},"reference_index":2,"source":"pdf_text","source_observed_at":"2026-08-11T12:47:23.940848Z"},"links":{"citing_paper":"/paper/2412.13990"},"observation_digest":"sha256:79844f07f336643e07fc86f65147a2c12e9eede49d7a3d9251bdea300e14d4d7","observation_id":"02960bcf-a510-453b-a62f-04aaa8dc85e5","resolution":{"observed_at":"2026-08-11T12:47:24.181012Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-16T06:30:59.297886+00:00","source":"crossref"},{"observed_at":"2026-08-16T06:30:54.164669+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-11T12:47:24.170693Z","title":"Characterization of optimization pro blems that are solv- able iteratively with linear convergence","venue":null,"work_id":"49a0dadf-92c0-455c-8de5-58ee6600c0de","year":2024},"citing_paper":{"arxiv_id":"2412.13990","last_updated":"2025-01-19T18:43:04Z","snapshot_observed_at":"2026-08-14T21:30:28.592183Z","submitted_at":"2024-12-18T16:10:44Z","title":"A geodesic convexity-like structure for the polar decomposition of a square matrix","version":2},"reference_index":3,"source":"pdf_text","source_observed_at":"2026-08-11T12:47:23.943538Z"},"links":{"citing_paper":"/paper/2412.13990"},"observation_digest":"sha256:7b69e0eef7074c1cf62ddd282fad18db04609704a400fbbc2b037229cbcef389","observation_id":"c448c792-1cc3-424d-a3e9-809fb68ed6d1","resolution":{"observed_at":"2026-08-11T12:47:24.173666Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-16T06:30:59.297886+00:00","source":"crossref"},{"observed_at":"2026-08-16T06:30:54.164669+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-11T12:47:24.162211Z","title":"Geodesic convex ity of the sym- metric eigenvalue problem and convergence of steepest desc ent","venue":null,"work_id":"35828402-0e9e-4605-873b-ba3de8e6e81f","year":2024},"citing_paper":{"arxiv_id":"2412.13990","last_updated":"2025-01-19T18:43:04Z","snapshot_observed_at":"2026-08-14T21:30:28.592183Z","submitted_at":"2024-12-18T16:10:44Z","title":"A geodesic convexity-like structure for the polar decomposition of a square matrix","version":2},"reference_index":4,"source":"pdf_text","source_observed_at":"2026-08-11T12:47:23.946487Z"},"links":{"citing_paper":"/paper/2412.13990"},"observation_digest":"sha256:9b39827812444581967b81a54e77dc7d9c6f132a74df5066038ab03c30624518","observation_id":"c35f7bb5-0eef-4e46-a592-bc9ff7fbad46","resolution":{"observed_at":"2026-08-11T12:47:24.165779Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-16T06:30:59.297886+00:00","source":"crossref"},{"observed_at":"2026-08-16T06:30:54.164669+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-11T12:47:24.154944Z","title":"Optimality conditions on the orthogona l group","venue":null,"work_id":"adad803c-d10f-46cd-9a45-bba781b43e6a","year":2023},"citing_paper":{"arxiv_id":"2412.13990","last_updated":"2025-01-19T18:43:04Z","snapshot_observed_at":"2026-08-14T21:30:28.592183Z","submitted_at":"2024-12-18T16:10:44Z","title":"A geodesic convexity-like structure for the polar decomposition of a square matrix","version":2},"reference_index":5,"source":"pdf_text","source_observed_at":"2026-08-11T12:47:23.949940Z"},"links":{"citing_paper":"/paper/2412.13990"},"observation_digest":"sha256:814a03fe0556fffaad4c83b19c9bdd1dbd570ee43bbf64f59bd051d027e2910c","observation_id":"5fc89825-79e1-42e6-a18d-ef72f77f94b1","resolution":{"observed_at":"2026-08-11T12:47:24.157787Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-16T06:30:59.297886+00:00","source":"crossref"},{"observed_at":"2026-08-16T06:30:54.164669+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":{"arxiv_id":"2006.08548","last_updated":"2020-06-15T17:05:08Z","snapshot_observed_at":"2026-08-16T09:42:36.304286Z","submitted_at":"2020-06-15T17:05:08Z","title":"A Note on Nesterov's Accelerated Method in Nonconvex Optimization: a Weak Estimate Sequence Approach","version":1},"cited_work":{"arxiv_id":"2006.08548","doi":null,"metadata_source":"pith","pith_arxiv_id":"2006.08548","snapshot_observed_at":"2026-08-11T12:47:24.039031Z","title":"A Note on Nesterov's Accelerated Method in Nonconvex Optimization: a Weak Estimate Sequence Approach","venue":"math.OC","work_id":"75b8d37f-9baf-4a68-a119-24fb2f13804e","year":2020},"citing_paper":{"arxiv_id":"2412.13990","last_updated":"2025-01-19T18:43:04Z","snapshot_observed_at":"2026-08-14T21:30:28.592183Z","submitted_at":"2024-12-18T16:10:44Z","title":"A geodesic convexity-like structure for the polar decomposition of a square matrix","version":2},"reference_index":6,"source":"pdf_text","source_observed_at":"2026-08-11T12:47:23.952653Z"},"links":{"cited_paper":"/paper/2006.08548","citing_paper":"/paper/2412.13990"},"observation_digest":"sha256:ae7c4c1953e4c1214e2244d1ea61bd6363fd4ceb6de2695579d7fa4e2999344e","observation_id":"0318ec4c-685d-4c88-89a1-bab81146b091","resolution":{"observed_at":"2026-08-11T12:47:24.042225Z","resolver_source":"local_arxiv","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-16T06:30:59.297886+00:00","source":"crossref"},{"observed_at":"2026-08-16T06:30:54.164669+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-11T12:47:24.147690Z","title":"A new scaling for newton’s ite ration for the polar decomposition and its backward stability","venue":null,"work_id":"899a855f-40b7-4845-b1f1-8f601c159aa5","year":2008},"citing_paper":{"arxiv_id":"2412.13990","last_updated":"2025-01-19T18:43:04Z","snapshot_observed_at":"2026-08-14T21:30:28.592183Z","submitted_at":"2024-12-18T16:10:44Z","title":"A geodesic convexity-like structure for the polar decomposition of a square matrix","version":2},"reference_index":7,"source":"pdf_text","source_observed_at":"2026-08-11T12:47:23.955851Z"},"links":{"citing_paper":"/paper/2412.13990"},"observation_digest":"sha256:4999100447bbdd0ac0f44ff688617a6404e5243fd249c2babcd9b3df3e6a6acb","observation_id":"1b866fbf-8b0a-4fa1-b651-0e02f04c0abb","resolution":{"observed_at":"2026-08-11T12:47:24.150623Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-16T06:30:59.297886+00:00","source":"crossref"},{"observed_at":"2026-08-16T06:30:54.164669+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-11T12:47:24.140381Z","title":"Error bounds, q uadratic growth, and linear convergence of proximal methods","venue":null,"work_id":"cb06ff7e-2abf-4370-810f-75f1dbda2667","year":2018},"citing_paper":{"arxiv_id":"2412.13990","last_updated":"2025-01-19T18:43:04Z","snapshot_observed_at":"2026-08-14T21:30:28.592183Z","submitted_at":"2024-12-18T16:10:44Z","title":"A geodesic convexity-like structure for the polar decomposition of a square matrix","version":2},"reference_index":8,"source":"pdf_text","source_observed_at":"2026-08-11T12:47:23.958455Z"},"links":{"citing_paper":"/paper/2412.13990"},"observation_digest":"sha256:bae29020f12680bc21c982eba51f62476319e991603cca89d40ef83c52aac97e","observation_id":"9aafd785-fd26-4929-a5e1-af2fbd7a7dcd","resolution":{"observed_at":"2026-08-11T12:47:24.143170Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-16T06:30:59.297886+00:00","source":"crossref"},{"observed_at":"2026-08-16T06:30:54.164669+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":{"arxiv_id":"2410.20040","last_updated":"2024-10-26T01:26:13Z","snapshot_observed_at":"2026-08-16T13:05:49.068823Z","submitted_at":"2024-10-26T01:26:13Z","title":"Studying Morphological Variation: Exploring the Shape Space in Evolutionary Anthropology","version":1},"cited_work":{"arxiv_id":"2410.20040","doi":null,"metadata_source":"pith","pith_arxiv_id":"2410.20040","snapshot_observed_at":"2026-08-11T12:47:24.028678Z","title":"Studying Morphological Variation: Exploring the Shape Space in Evolutionary Anthropology","venue":"math.NA","work_id":"da4f7ece-621c-40b7-9362-62b73edd9ba7","year":2024},"citing_paper":{"arxiv_id":"2412.13990","last_updated":"2025-01-19T18:43:04Z","snapshot_observed_at":"2026-08-14T21:30:28.592183Z","submitted_at":"2024-12-18T16:10:44Z","title":"A geodesic convexity-like structure for the polar decomposition of a square matrix","version":2},"reference_index":9,"source":"pdf_text","source_observed_at":"2026-08-11T12:47:23.960982Z"},"links":{"cited_paper":"/paper/2410.20040","citing_paper":"/paper/2412.13990"},"observation_digest":"sha256:e9339994c92aeb0fe238460ed8a8a436c62b091ff4c4fd40de5be8b00adb2acb","observation_id":"fe0c0231-2be1-4acc-bfa8-3f61897d3548","resolution":{"observed_at":"2026-08-11T12:47:24.031955Z","resolver_source":"local_arxiv","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-16T06:30:59.297886+00:00","source":"crossref"},{"observed_at":"2026-08-16T06:30:54.164669+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-11T12:47:24.132710Z","title":"On halley’s iteration method","venue":null,"work_id":"e650d4bf-3259-4c87-aacd-8515dc7f81bb","year":1985},"citing_paper":{"arxiv_id":"2412.13990","last_updated":"2025-01-19T18:43:04Z","snapshot_observed_at":"2026-08-14T21:30:28.592183Z","submitted_at":"2024-12-18T16:10:44Z","title":"A geodesic convexity-like structure for the polar decomposition of a square matrix","version":2},"reference_index":10,"source":"pdf_text","source_observed_at":"2026-08-11T12:47:23.964161Z"},"links":{"citing_paper":"/paper/2412.13990"},"observation_digest":"sha256:b29946824b5faccfa251aaedd625d102e08d1bdc9f6ef59833dca29db0acf403","observation_id":"ffc504c2-b70f-4839-89dc-0617982b4bcd","resolution":{"observed_at":"2026-08-11T12:47:24.135743Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-16T06:30:59.297886+00:00","source":"crossref"},{"observed_at":"2026-08-16T06:30:54.164669+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":{"arxiv_id":"1710.00797","last_updated":"2023-05-16T11:26:49Z","snapshot_observed_at":"2026-08-15T23:21:02.466065Z","submitted_at":"2017-10-02T17:10:43Z","title":"Accelerated Methods for $\\alpha$-Weakly-Quasi-Convex Problems","version":7},"cited_work":{"arxiv_id":"1710.00797","doi":null,"metadata_source":"pith","pith_arxiv_id":"1710.00797","snapshot_observed_at":"2026-08-11T12:47:24.016104Z","title":"Accelerated Methods for $\\alpha$-Weakly-Quasi-Convex Problems","venue":"math.OC","work_id":"fc53397c-b0ab-4968-b838-161133a616fd","year":2017},"citing_paper":{"arxiv_id":"2412.13990","last_updated":"2025-01-19T18:43:04Z","snapshot_observed_at":"2026-08-14T21:30:28.592183Z","submitted_at":"2024-12-18T16:10:44Z","title":"A geodesic convexity-like structure for the polar decomposition of a square matrix","version":2},"reference_index":11,"source":"pdf_text","source_observed_at":"2026-08-11T12:47:23.966673Z"},"links":{"cited_paper":"/paper/1710.00797","citing_paper":"/paper/2412.13990"},"observation_digest":"sha256:bc4bd1dbe62442677a5f43c14afc38c9e7018f822f06faddbc16f78b931a7698","observation_id":"aade470f-826f-4654-84d4-a39eb664d61a","resolution":{"observed_at":"2026-08-11T12:47:24.020859Z","resolver_source":"local_arxiv","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-16T06:30:59.297886+00:00","source":"crossref"},{"observed_at":"2026-08-16T06:30:54.164669+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-11T12:47:24.125292Z","title":"Gradient d escent learns linear dynamical systems","venue":null,"work_id":"1391f1ec-f64a-42fc-afc0-83b54f49e804","year":2018},"citing_paper":{"arxiv_id":"2412.13990","last_updated":"2025-01-19T18:43:04Z","snapshot_observed_at":"2026-08-14T21:30:28.592183Z","submitted_at":"2024-12-18T16:10:44Z","title":"A geodesic convexity-like structure for the polar decomposition of a square matrix","version":2},"reference_index":12,"source":"pdf_text","source_observed_at":"2026-08-11T12:47:23.969742Z"},"links":{"citing_paper":"/paper/2412.13990"},"observation_digest":"sha256:360aef0f38c45afbcbc39ddecdcba635b9013640c35327c4775cce163cc4f741","observation_id":"648a9515-b5c8-44d0-bf4f-d1a55ed6d484","resolution":{"observed_at":"2026-08-11T12:47:24.128051Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-16T06:30:59.297886+00:00","source":"crossref"},{"observed_at":"2026-08-16T06:30:54.164669+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-11T12:47:24.117587Z","title":"Computing the polar decomposition—w ith applica- tions","venue":null,"work_id":"4f66fa2b-d1fd-499e-9449-57cb6801236a","year":1986},"citing_paper":{"arxiv_id":"2412.13990","last_updated":"2025-01-19T18:43:04Z","snapshot_observed_at":"2026-08-14T21:30:28.592183Z","submitted_at":"2024-12-18T16:10:44Z","title":"A geodesic convexity-like structure for the polar decomposition of a square matrix","version":2},"reference_index":13,"source":"pdf_text","source_observed_at":"2026-08-11T12:47:23.972431Z"},"links":{"citing_paper":"/paper/2412.13990"},"observation_digest":"sha256:58551c5076401589afa0f573b76ad7bde2d85505cecb4fc2b14b56939b73841b","observation_id":"34275b6f-00bb-46fc-be40-4a5d33277292","resolution":{"observed_at":"2026-08-11T12:47:24.120484Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-16T06:30:59.297886+00:00","source":"crossref"},{"observed_at":"2026-08-16T06:30:54.164669+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-11T12:47:24.109984Z","title":"Functions of matrices: Theory and co mputation, 2008","venue":null,"work_id":"375a8a06-e73e-44b2-8001-3c6fa1a386a5","year":2008},"citing_paper":{"arxiv_id":"2412.13990","last_updated":"2025-01-19T18:43:04Z","snapshot_observed_at":"2026-08-14T21:30:28.592183Z","submitted_at":"2024-12-18T16:10:44Z","title":"A geodesic convexity-like structure for the polar decomposition of a square matrix","version":2},"reference_index":14,"source":"pdf_text","source_observed_at":"2026-08-11T12:47:23.975352Z"},"links":{"citing_paper":"/paper/2412.13990"},"observation_digest":"sha256:3132e38d37111f24381a9e1e951d33768166665d641829425e588edc49359894","observation_id":"d60f8d75-4839-4ebb-8408-7b964ed8673a","resolution":{"observed_at":"2026-08-11T12:47:24.112973Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-16T06:30:59.297886+00:00","source":"crossref"},{"observed_at":"2026-08-16T06:30:54.164669+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-11T12:47:24.102247Z","title":"A solution for the best rotation to rel ate two sets of vectors","venue":null,"work_id":"d3a5108e-1ee9-4704-9bd8-eaf15514911e","year":1976},"citing_paper":{"arxiv_id":"2412.13990","last_updated":"2025-01-19T18:43:04Z","snapshot_observed_at":"2026-08-14T21:30:28.592183Z","submitted_at":"2024-12-18T16:10:44Z","title":"A geodesic convexity-like structure for the polar decomposition of a square matrix","version":2},"reference_index":15,"source":"pdf_text","source_observed_at":"2026-08-11T12:47:23.978179Z"},"links":{"citing_paper":"/paper/2412.13990"},"observation_digest":"sha256:e04b038f47104986fd5568c62e2aa9b03dfee1d5effb461805fe3342caac659e","observation_id":"4f989b37-002b-4074-84b1-13fbdf51982e","resolution":{"observed_at":"2026-08-11T12:47:24.105044Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-16T06:30:59.297886+00:00","source":"crossref"},{"observed_at":"2026-08-16T06:30:54.164669+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-11T12:47:24.094602Z","title":"Linear co nvergence of gradient and proximal-gradient methods under the polyak-ł ojasiewicz con- dition","venue":null,"work_id":"0437ead6-e376-46b2-a146-82caa41b821d","year":2016},"citing_paper":{"arxiv_id":"2412.13990","last_updated":"2025-01-19T18:43:04Z","snapshot_observed_at":"2026-08-14T21:30:28.592183Z","submitted_at":"2024-12-18T16:10:44Z","title":"A geodesic convexity-like structure for the polar decomposition of a square matrix","version":2},"reference_index":16,"source":"pdf_text","source_observed_at":"2026-08-11T12:47:23.980917Z"},"links":{"citing_paper":"/paper/2412.13990"},"observation_digest":"sha256:fd6ba8207383166bb5b9f266ca9d4b5e99e0fcf9e80ee0664a28a168292d82c8","observation_id":"19aac3e2-79f5-4ed5-b351-6cd3fad10506","resolution":{"observed_at":"2026-08-11T12:47:24.097747Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-16T06:30:59.297886+00:00","source":"crossref"},{"observed_at":"2026-08-16T06:30:54.164669+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-11T12:47:24.087103Z","title":null,"venue":null,"work_id":"2e06234c-3ffd-40ea-8c59-ed0919ef65df","year":1990},"citing_paper":{"arxiv_id":"2412.13990","last_updated":"2025-01-19T18:43:04Z","snapshot_observed_at":"2026-08-14T21:30:28.592183Z","submitted_at":"2024-12-18T16:10:44Z","title":"A geodesic convexity-like structure for the polar decomposition of a square matrix","version":2},"reference_index":17,"source":"pdf_text","source_observed_at":"2026-08-11T12:47:23.983458Z"},"links":{"citing_paper":"/paper/2412.13990"},"observation_digest":"sha256:ea106ea7f17b233d436bb3cc2a18cda96c6f16437e588a0e99e9a063c3a19487","observation_id":"df25e070-c6ee-4479-8911-273a1f1d7bff","resolution":{"observed_at":"2026-08-11T12:47:24.089979Z","resolver_source":"raw_fallback","status":"unresolved"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-16T06:30:59.297886+00:00","source":"crossref"},{"observed_at":"2026-08-16T06:30:54.164669+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-11T12:47:24.079138Z","title":"Opti mizing halley’s iteration for computing the matrix polar decomposition","venue":null,"work_id":"e4e4f9c6-d241-44f2-8c6f-d307e48da668","year":2010},"citing_paper":{"arxiv_id":"2412.13990","last_updated":"2025-01-19T18:43:04Z","snapshot_observed_at":"2026-08-14T21:30:28.592183Z","submitted_at":"2024-12-18T16:10:44Z","title":"A geodesic convexity-like structure for the polar decomposition of a square matrix","version":2},"reference_index":18,"source":"pdf_text","source_observed_at":"2026-08-11T12:47:23.986094Z"},"links":{"citing_paper":"/paper/2412.13990"},"observation_digest":"sha256:076145cc0551721645dfebf87486f4dce3c11b1700a7f983291bfbfb7e5dc64e","observation_id":"62ab2cc5-78c8-498f-b1c2-2c6d7bf3f728","resolution":{"observed_at":"2026-08-11T12:47:24.082238Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-16T06:30:59.297886+00:00","source":"crossref"},{"observed_at":"2026-08-16T06:30:54.164669+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-11T12:47:24.071453Z","title":"Backward stabi lity of iterations for computing the polar decomposition","venue":null,"work_id":"f1b502d0-491d-41a8-8ba5-b8dc7a281cc6","year":2012},"citing_paper":{"arxiv_id":"2412.13990","last_updated":"2025-01-19T18:43:04Z","snapshot_observed_at":"2026-08-14T21:30:28.592183Z","submitted_at":"2024-12-18T16:10:44Z","title":"A geodesic convexity-like structure for the polar decomposition of a square matrix","version":2},"reference_index":19,"source":"pdf_text","source_observed_at":"2026-08-11T12:47:23.988688Z"},"links":{"citing_paper":"/paper/2412.13990"},"observation_digest":"sha256:28217e6bf9009e57ba0ca0c8b00ec768fc7716f413655b9722694cc75efb7e13","observation_id":"eb3a5ae3-b580-4863-a35e-6ff277dab1ce","resolution":{"observed_at":"2026-08-11T12:47:24.074291Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-16T06:30:59.297886+00:00","source":"crossref"},{"observed_at":"2026-08-16T06:30:54.164669+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-11T12:47:24.063284Z","title":"Lin ear convergence of first order methods for non-strongly convex optimization","venue":null,"work_id":"37cdd535-9ef2-4bfa-a285-3be540084e57","year":2019},"citing_paper":{"arxiv_id":"2412.13990","last_updated":"2025-01-19T18:43:04Z","snapshot_observed_at":"2026-08-14T21:30:28.592183Z","submitted_at":"2024-12-18T16:10:44Z","title":"A geodesic convexity-like structure for the polar decomposition of a square matrix","version":2},"reference_index":20,"source":"pdf_text","source_observed_at":"2026-08-11T12:47:23.991299Z"},"links":{"citing_paper":"/paper/2412.13990"},"observation_digest":"sha256:e810366169bc1af632a0d54ae00d639538eef9150d341f65d1c10b691c3a3a19","observation_id":"ae105528-bdb6-4c60-bd2c-0242d35f1c97","resolution":{"observed_at":"2026-08-11T12:47:24.066299Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-16T06:30:59.297886+00:00","source":"crossref"},{"observed_at":"2026-08-16T06:30:54.164669+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-11T12:47:24.055576Z","title":"Introduction to Differential Geometry","venue":null,"work_id":"17b9aa96-3d35-49e9-8985-3eaee78e57b7","year":2022},"citing_paper":{"arxiv_id":"2412.13990","last_updated":"2025-01-19T18:43:04Z","snapshot_observed_at":"2026-08-14T21:30:28.592183Z","submitted_at":"2024-12-18T16:10:44Z","title":"A geodesic convexity-like structure for the polar decomposition of a square matrix","version":2},"reference_index":21,"source":"pdf_text","source_observed_at":"2026-08-11T12:47:23.993970Z"},"links":{"citing_paper":"/paper/2412.13990"},"observation_digest":"sha256:159df0edc49ccbfd3f86638d2c12dc00698d4ce670132aaa1b995f7a84834102","observation_id":"23884369-e6b6-48e5-a0df-db7f55ea588f","resolution":{"observed_at":"2026-08-11T12:47:24.058296Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-16T06:30:59.297886+00:00","source":"crossref"},{"observed_at":"2026-08-16T06:30:54.164669+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-11T12:47:24.047756Z","title":"A generalized solution of the ortho gonal procrustes problem","venue":null,"work_id":"4421b58b-7098-4f9f-b406-95b41240da08","year":1966},"citing_paper":{"arxiv_id":"2412.13990","last_updated":"2025-01-19T18:43:04Z","snapshot_observed_at":"2026-08-14T21:30:28.592183Z","submitted_at":"2024-12-18T16:10:44Z","title":"A geodesic convexity-like structure for the polar decomposition of a square matrix","version":2},"reference_index":22,"source":"pdf_text","source_observed_at":"2026-08-11T12:47:23.996654Z"},"links":{"citing_paper":"/paper/2412.13990"},"observation_digest":"sha256:7b6bbfd3c8c44664606a4515ba2904dd0e06d4e7b583a8aa4d7b6c74ae09b4b4","observation_id":"01c70155-8532-4d19-b438-e4759db0d57f","resolution":{"observed_at":"2026-08-11T12:47:24.050612Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-16T06:30:59.297886+00:00","source":"crossref"},{"observed_at":"2026-08-16T06:30:54.164669+00:00","source":"retraction_watch"}],"state":"measured"}}],"paper":{"arxiv_id":"2412.13990","last_updated":"2025-01-19T18:43:04Z","latest_version":2,"primary_category":"math.NA","snapshot_observed_at":"2026-08-14T21:30:28.592183Z","submitted_at":"2024-12-18T16:10:44Z","title":"A geodesic convexity-like structure for the polar decomposition of a square matrix"},"reference_resolution":{"displayed":22,"state_counts":{"malformed_identifier":0,"metadata_mismatch":0,"parse_uncertain":0,"unresolved":1,"verified_exact":3,"verified_fuzzy":18},"total_outbound_references":22},"refusal":"A citation records a reference. It does not transfer a finding from one paper to another.","schema":"pith.paper-citation-record.v1","standing_sources":[{"observed_at":"2026-08-16T06:30:59.297886+00:00","source":"crossref"},{"observed_at":"2026-08-16T06:30:54.164669+00:00","source":"retraction_watch"}],"thesis":"As of 16 August 2026, this Paper Citation Record lists 22 of 22 outbound references and 0 inbound Pith citation observations for arXiv:2412.13990."}