{"as_of":"2026-08-22T06:47:00Z","caps":{"database_statements":6,"inbound":100,"outbound":100},"context_digest":"sha256:0608a8b2012b94a13d9163a1e2e8a7d9903145e69a2423099e028057809c849d","coverage":[{"denominator":99,"lane":"reference_resolution","note":"Typed states for the displayed outbound observations.","records_observed":99,"source":"paper_references, paper_reference_links","source_observed_at":"2026-08-16T04:56:23.171892Z","state":"measured"},{"denominator":109,"lane":"standing_notices","note":"One-hop event checks from named stored sources.","records_observed":109,"source":"scholarly_work_events, retraction_status_cache","source_observed_at":"2026-08-22T06:32:14.747728+00:00","state":"measured"},{"denominator":10,"lane":"inbound_itemization","note":"Pith citing papers itemized under the disclosed page cap.","records_observed":10,"source":"paper_references, paper_reference_links","source_observed_at":"2026-08-15T15:14:01.456811Z","state":"measured"},{"denominator":1,"lane":"external_citation_measurements","note":"A source-named dated measurement, never combined with another source.","records_observed":1,"source":"arxiv_reference","source_observed_at":"2026-08-05T02:28:24.338817Z","state":"measured"}],"external_citation_measurements":[{"count":0,"observed_at":"2026-08-05T02:28:24.338817Z","source":"arxiv_reference"}],"inbound":[{"citation":{"cited_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"cited_work":{"arxiv_id":"2505.00351","doi":"10.48550/arxiv.2505.00351","metadata_source":"arxiv_reference","pith_arxiv_id":"2505.00351","snapshot_observed_at":"2026-08-05T02:28:24.338817Z","title":"Integral representations of S obolev spaces via ReLU k activation function and optimal error estimates for linearized networks","venue":"ArXiv.org","work_id":"965d9dd8-295d-4e84-b5b1-b4b9c294d900","year":2025},"citing_paper":{"arxiv_id":"2602.14757","last_updated":"2026-04-17T05:22:28Z","snapshot_observed_at":"2026-08-14T01:22:09.530473Z","submitted_at":"2026-02-16T14:01:50Z","title":"Solving Inverse Parametrized Problems via Finite Elements and Extreme Learning Networks","version":2},"reference_index":37,"source":"pdf_text","source_observed_at":"2026-05-15T21:57:48.536230Z"},"links":{"cited_paper":"/paper/2505.00351","citing_paper":"/paper/2602.14757"},"observation_digest":"sha256:b3725f83e08271750a24672f8b926430628449ea436507aaa4fe1113a13ce0ba","observation_id":"9bac93bc-c39a-411b-abb8-949a35bb4f6e","resolution":{"observed_at":"2026-05-15T22:00:20.890588Z","resolver_source":"arxiv_id","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-22T06:32:14.747728+00:00","source":"crossref"},{"observed_at":"2026-08-22T06:32:06.552537+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"cited_work":{"arxiv_id":"2505.00351","doi":"10.48550/arxiv.2505.00351","metadata_source":"arxiv_reference","pith_arxiv_id":"2505.00351","snapshot_observed_at":"2026-08-05T02:28:24.338817Z","title":"Integral representations of S obolev spaces via ReLU k activation function and optimal error estimates for linearized networks","venue":"ArXiv.org","work_id":"965d9dd8-295d-4e84-b5b1-b4b9c294d900","year":2025},"citing_paper":{"arxiv_id":"2604.08869","last_updated":"2026-08-15T05:34:57Z","snapshot_observed_at":"2026-08-20T23:09:27.540038Z","submitted_at":"2026-04-10T02:07:07Z","title":"Adaptive Randomized Neural Networks with Locally Activation Function: Theory and Algorithm for Solving PDEs","version":1},"reference_index":25,"source":"pdf_text","source_observed_at":"2026-05-10T18:00:14.153569Z"},"links":{"cited_paper":"/paper/2505.00351","citing_paper":"/paper/2604.08869"},"observation_digest":"sha256:111ae3d9dc16af522caae07b27e456cbf730d4ec398a8b324d30a4b07021b0ec","observation_id":"634f9e38-9802-44ff-8824-b09a61b69d8d","resolution":{"observed_at":"2026-05-11T05:40:59.971219Z","resolver_source":"arxiv_id","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-22T06:32:14.747728+00:00","source":"crossref"},{"observed_at":"2026-08-22T06:32:06.552537+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"cited_work":{"arxiv_id":"2505.00351","doi":"10.48550/arxiv.2505.00351","metadata_source":"arxiv_reference","pith_arxiv_id":"2505.00351","snapshot_observed_at":"2026-08-05T02:28:24.338817Z","title":"Integral representations of S obolev spaces via ReLU k activation function and optimal error estimates for linearized networks","venue":"ArXiv.org","work_id":"965d9dd8-295d-4e84-b5b1-b4b9c294d900","year":2025},"citing_paper":{"arxiv_id":"2605.17692","last_updated":"2026-05-17T23:20:50Z","snapshot_observed_at":"2026-08-16T13:09:07.775928Z","submitted_at":"2026-05-17T23:20:50Z","title":"Exact Convex Reformulations of Linear Neural Networks via Completely Positive Lifting","version":1},"reference_index":17,"source":"arxiv_source","source_observed_at":"2026-05-20T13:36:05.373250Z"},"links":{"cited_paper":"/paper/2505.00351","citing_paper":"/paper/2605.17692"},"observation_digest":"sha256:f30a1cc872edf4df1b1ee22cea375c112a10055b3363c3a5a64d095081ed8f36","observation_id":"68966dfb-20d2-4b5b-b141-551349ffa8d5","resolution":{"observed_at":"2026-05-20T13:38:19.309845Z","resolver_source":"arxiv_id","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-22T06:32:14.747728+00:00","source":"crossref"},{"observed_at":"2026-08-22T06:32:06.552537+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"cited_work":{"arxiv_id":"2505.00351","doi":"10.48550/arxiv.2505.00351","metadata_source":"arxiv_reference","pith_arxiv_id":"2505.00351","snapshot_observed_at":"2026-08-05T02:28:24.338817Z","title":"Integral representations of S obolev spaces via ReLU k activation function and optimal error estimates for linearized networks","venue":"ArXiv.org","work_id":"965d9dd8-295d-4e84-b5b1-b4b9c294d900","year":2025},"citing_paper":{"arxiv_id":"2605.22557","last_updated":"2026-05-26T09:40:19Z","snapshot_observed_at":"2026-08-18T16:44:52.398610Z","submitted_at":"2026-05-21T14:39:43Z","title":"Neural Flow Operators can Approximate any Operator: Abstract Frameworks and Universal Approximations","version":1},"reference_index":150,"source":"arxiv_source","source_observed_at":"2026-05-22T07:28:49.152516Z"},"links":{"cited_paper":"/paper/2505.00351","citing_paper":"/paper/2605.22557"},"observation_digest":"sha256:6ae56576f529e9f82ad76d2b3c202d856565c599495d4e9bc6995bedd8a5b94a","observation_id":"d12785ea-5e5b-425e-860b-44dfb252ae19","resolution":{"observed_at":"2026-05-22T07:31:14.027898Z","resolver_source":"arxiv_id","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-22T06:32:14.747728+00:00","source":"crossref"},{"observed_at":"2026-08-22T06:32:06.552537+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"cited_work":{"arxiv_id":"2505.00351","doi":"10.48550/arxiv.2505.00351","metadata_source":"arxiv_reference","pith_arxiv_id":"2505.00351","snapshot_observed_at":"2026-08-05T02:28:24.338817Z","title":"Integral representations of S obolev spaces via ReLU k activation function and optimal error estimates for linearized networks","venue":"ArXiv.org","work_id":"965d9dd8-295d-4e84-b5b1-b4b9c294d900","year":2025},"citing_paper":{"arxiv_id":"2605.22557","last_updated":"2026-05-26T09:40:19Z","snapshot_observed_at":"2026-08-18T16:44:52.398610Z","submitted_at":"2026-05-21T14:39:43Z","title":"Neural Flow Operators can Approximate any Operator: Abstract Frameworks and Universal Approximations","version":2},"reference_index":54,"source":"arxiv_source","source_observed_at":"2026-06-30T17:08:54.066574Z"},"links":{"cited_paper":"/paper/2505.00351","citing_paper":"/paper/2605.22557"},"observation_digest":"sha256:0a1b85e40ada7715fa99914088e859ac742f199f2eecee06698d738290fcde02","observation_id":"db603299-b28c-4a12-8c2e-326c47adc6f1","resolution":{"observed_at":"2026-06-30T17:14:57.095714Z","resolver_source":"arxiv_id","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-22T06:32:14.747728+00:00","source":"crossref"},{"observed_at":"2026-08-22T06:32:06.552537+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"cited_work":{"arxiv_id":"2505.00351","doi":"10.48550/arxiv.2505.00351","metadata_source":"arxiv_reference","pith_arxiv_id":"2505.00351","snapshot_observed_at":"2026-08-05T02:28:24.338817Z","title":"Integral representations of S obolev spaces via ReLU k activation function and optimal error estimates for linearized networks","venue":"ArXiv.org","work_id":"965d9dd8-295d-4e84-b5b1-b4b9c294d900","year":2025},"citing_paper":{"arxiv_id":"2606.24795","last_updated":"2026-06-23T16:48:30Z","snapshot_observed_at":"2026-08-17T14:41:11.991541Z","submitted_at":"2026-06-23T16:48:30Z","title":"Sharp Sobolev Sandwich and Approximation Rates of Radon-Domain $L^p$ Ridge Integral Spaces for ReLU$^k$ Networks","version":1},"reference_index":31,"source":"pdf_text","source_observed_at":"2026-06-25T22:43:51.421346Z"},"links":{"cited_paper":"/paper/2505.00351","citing_paper":"/paper/2606.24795"},"observation_digest":"sha256:1647ee2d64d7e93d24dfada38dcd0bd754a196109cebb75bf239352529795841","observation_id":"cb7fde86-4d4a-4fd1-b149-edc860b60e79","resolution":{"observed_at":"2026-07-04T18:40:02.563295Z","resolver_source":"arxiv_id","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-22T06:32:14.747728+00:00","source":"crossref"},{"observed_at":"2026-08-22T06:32:06.552537+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"cited_work":{"arxiv_id":"2505.00351","doi":"10.48550/arxiv.2505.00351","metadata_source":"arxiv_reference","pith_arxiv_id":"2505.00351","snapshot_observed_at":"2026-08-05T02:28:24.338817Z","title":"Integral representations of S obolev spaces via ReLU k activation function and optimal error estimates for linearized networks","venue":"ArXiv.org","work_id":"965d9dd8-295d-4e84-b5b1-b4b9c294d900","year":2025},"citing_paper":{"arxiv_id":"2606.31342","last_updated":"2026-06-30T08:41:54Z","snapshot_observed_at":"2026-08-07T20:18:17.107324Z","submitted_at":"2026-06-30T08:41:54Z","title":"Domain-Decomposed Randomized Neural Networks for Partial Differential Equations in Unbounded Domains","version":1},"reference_index":22,"source":"pdf_text","source_observed_at":"2026-07-01T04:47:48.362228Z"},"links":{"cited_paper":"/paper/2505.00351","citing_paper":"/paper/2606.31342"},"observation_digest":"sha256:2c9d636757b0106852357e5fed00ef7608fb7558eccd824bdcfaf72f28d96dd9","observation_id":"befc5516-450a-4c63-8a59-69cd296033a6","resolution":{"observed_at":"2026-07-01T11:05:41.681859Z","resolver_source":"arxiv_id","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-22T06:32:14.747728+00:00","source":"crossref"},{"observed_at":"2026-08-22T06:32:06.552537+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2505.00351","snapshot_observed_at":"2026-08-01T04:44:44.166201Z","title":"arXiv preprint arXiv:2505.00351 (2025)","venue":null,"work_id":null,"year":2025},"citing_paper":{"arxiv_id":"2607.22478","last_updated":"2026-07-24T16:47:05Z","snapshot_observed_at":"2026-08-21T00:41:08.383872Z","submitted_at":"2026-07-24T16:47:05Z","title":"ReLU$^k$ Neural de Rham Complexes","version":1},"reference_index":20,"source":"pdf_text","source_observed_at":"2026-08-01T04:44:44.166201Z"},"links":{"cited_paper":"/paper/2505.00351","citing_paper":"/paper/2607.22478"},"observation_digest":"sha256:b07aa7cbc7bf2926c07d10312dfd3786b797e0a5beeeab9e8caeeab1ef0dd3b1","observation_id":"667c7e0a-d018-4739-a550-f63b874cfcc8","resolution":{"observed_at":"2026-08-01T04:44:44.166201Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2505.00351","snapshot_observed_at":"2026-08-15T15:14:01.456811Z","title":null,"venue":null,"work_id":null,"year":2025},"citing_paper":{"arxiv_id":"2608.01357","last_updated":"2026-08-02T16:20:19Z","snapshot_observed_at":"2026-08-18T18:12:52.227332Z","submitted_at":"2026-08-02T16:20:19Z","title":"Do Neural Networks Really Beat the Curse of Dimensionality? A Bit-Complexity View","version":1},"reference_index":38,"source":"pdf_text","source_observed_at":"2026-08-15T15:14:01.456811Z"},"links":{"cited_paper":"/paper/2505.00351","citing_paper":"/paper/2608.01357"},"observation_digest":"sha256:78f5a46adcc3545718e83df2f0b591ea8cc3a5a5783b6fbbf57b2c94360fadd4","observation_id":"c5ea7eb1-6caf-400b-bef6-48cb52c7f067","resolution":{"observed_at":"2026-08-15T15:14:01.456811Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2505.00351","snapshot_observed_at":"2026-08-10T22:36:38.483855Z","title":null,"venue":null,"work_id":null,"year":2025},"citing_paper":{"arxiv_id":"2608.06687","last_updated":"2026-08-07T01:22:27Z","snapshot_observed_at":"2026-08-20T07:05:16.760115Z","submitted_at":"2026-08-07T01:22:27Z","title":"Optimal Neural Network Approximation via Empirical Least Squares with Deterministic Samples","version":1},"reference_index":49,"source":"pdf_text","source_observed_at":"2026-08-10T22:36:38.483855Z"},"links":{"cited_paper":"/paper/2505.00351","citing_paper":"/paper/2608.06687"},"observation_digest":"sha256:252020a82e6174ea84b5f30efb21a08833ff56db0cc987eeddb12b766c176073","observation_id":"7bbd1dcd-6000-40cc-a7e6-51bd5e5231d2","resolution":{"observed_at":"2026-08-10T22:36:38.483855Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}}],"links":{"evidence":"/evidence","html":"/paper/2505.00351/citation-record","integrity":"/paper/2505.00351/integrity","json":"/paper/2505.00351/citation-record.json","paper":"/paper/2505.00351"},"outbound":[{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-16T04:56:22.214421Z","title":"Support vector machines","venue":null,"work_id":null,"year":2008},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":1,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:22.214421Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:c3e886152583d2308789f8babbcb6712f7a678f1429c2f09c49ec4b8321725ab","observation_id":"6a2169db-8dce-487a-a802-2ae316b0f41b","resolution":{"observed_at":"2026-08-16T04:56:22.214421Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-16T04:56:22.224254Z","title":"Breaking the curse of dimensionality with convex ne ural networks","venue":null,"work_id":null,"year":2017},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":2,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:22.224254Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:a84b4c72f84691b2bad6cf2a6e47e1c7cc67ff811c362f672a636cd51e16b3d6","observation_id":"46fe8b5f-3272-45e9-8fea-165211f33e08","resolution":{"observed_at":"2026-08-16T04:56:22.224254Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-16T04:56:22.239828Z","title":"On the equivalence between kernel quadrature r ules and random feature ex- pansions","venue":null,"work_id":null,"year":2017},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":3,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:22.239828Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:4d8db3842b7941c47580287b42485a9889f82a09f70e1754110fc35120379637","observation_id":"9860d3f5-12a5-4fb1-8139-73821ca866df","resolution":{"observed_at":"2026-08-16T04:56:22.239828Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-16T04:56:22.256080Z","title":"Universal approximation bounds for superpo sitions of a sigmoidal function","venue":null,"work_id":null,"year":1993},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":4,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:22.256080Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:9bc9c5276322d6cf63a7d3265d7a79cdbe19a9b7a686560ce64012f5dced165c","observation_id":"f54ce81e-b75f-4a6b-851a-9f08120c8360","resolution":{"observed_at":"2026-08-16T04:56:22.256080Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-16T04:56:22.274042Z","title":"Approximation and estimation bounds for artiﬁ cial neural networks","venue":null,"work_id":null,"year":1994},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":5,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:22.274042Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:986c4c64ed21a80fe4f4415da0e48e43d08456cc14d594647807fc059f382c99","observation_id":"469e300a-0f3d-45b3-9381-47bf5e72353d","resolution":{"observed_at":"2026-08-16T04:56:22.274042Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-16T04:56:22.280850Z","title":"Approximation and learning by greedy algorithms","venue":null,"work_id":null,"year":2008},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":6,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:22.280850Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:da0779f8624d3abd607ec11e496ad3813a4baa9ec1fefe623525066824026322","observation_id":"51f97681-8d4f-4e1a-b4f4-13803f731e77","resolution":{"observed_at":"2026-08-16T04:56:22.280850Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-16T04:56:22.291692Z","title":"Nearly-tight vc- dimension and pseudodimension bounds for piecewise linear neural ne tworks","venue":null,"work_id":null,"year":2019},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":7,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:22.291692Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:3416f97f00a75b3b6661175c03c80243d7634e3b60c1dace50cd46df5d138817","observation_id":"d177910f-2d2e-4661-b319-175f17ac8b40","resolution":{"observed_at":"2026-08-16T04:56:22.291692Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-16T04:56:22.301874Z","title":"Bartlett and Shahar Mendelson","venue":null,"work_id":null,"year":2002},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":8,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:22.301874Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:4c829c6478a3f1e241d4e7da3305e187db7c84fc32644d980e9903e733fdea37","observation_id":"0df54708-9e65-4104-a18c-e9b6de4140e5","resolution":{"observed_at":"2026-08-16T04:56:22.301874Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-16T04:56:22.324507Z","title":"Two models of double descent for weak features","venue":null,"work_id":null,"year":2020},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":9,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:22.324507Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:5cb1dc0097f7b8b2338982f890b06d29c9c02e432fe0d597c65ae68f653881ef","observation_id":"4e1c1a01-1b90-4a24-a9f4-8e493c571c7a","resolution":{"observed_at":"2026-08-16T04:56:22.324507Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-16T04:56:22.331272Z","title":"Springer Science & Business Media, 2012","venue":null,"work_id":null,"year":2012},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":10,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:22.331272Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:79c7691812318da260e286c433573b1191833c3a99389133b2b80e28076b5bfe","observation_id":"5d030a25-f1f1-46fa-8fab-d6b3a03ea0b9","resolution":{"observed_at":"2026-08-16T04:56:22.331272Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-16T04:56:22.342944Z","title":"Optimal asymptotic bounds for spherical designs","venue":null,"work_id":null,"year":2013},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":11,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:22.342944Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:0734e2ffc09817c415b3cb8b656dd3edbc4ff4031c6f6a8da9a673d50383de61","observation_id":"0e5fa630-c0db-42a8-a6be-42ed29ae31be","resolution":{"observed_at":"2026-08-16T04:56:22.342944Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-16T04:56:22.365542Z","title":"C oncentration inequalities","venue":null,"work_id":null,"year":2003},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":12,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:22.365542Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:1a851a48bdf95cec156f4666404d721529905822daff857c18401c9f6d2b80fe","observation_id":"fc801d29-f183-40f2-b15f-ef0fbf8b1861","resolution":{"observed_at":"2026-08-16T04:56:22.365542Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-16T04:56:22.374666Z","title":"Projection bodies","venue":null,"work_id":null,"year":1986},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":13,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:22.374666Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:0f6ef19dedafdb7d9521733a20ac988858eb5f408d2039a86fa82658d34f488a","observation_id":"8a5af615-a8fe-4e0b-867e-e93c182e8b93","resolution":{"observed_at":"2026-08-16T04:56:22.374666Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-16T04:56:22.381073Z","title":"Weak type estimates for Cesaro sums of Jacobi polynomial series , volume 487","venue":null,"work_id":null,"year":1993},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":14,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:22.381073Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:810f1da40134cb07266e61c2b4fc3319f5ef6278df9b1c69b84d374e39825585","observation_id":"aba87ce3-2e52-4aab-a8d1-e8b54f57d49e","resolution":{"observed_at":"2026-08-16T04:56:22.381073Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-16T04:56:22.390432Z","title":"Bridgin g traditional and machine learning-based algorithms for solving pdes: the random feature me thod","venue":null,"work_id":null,"year":2022},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":15,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:22.390432Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:ffe5e3a2c2d5464705e89167b88915ba415f7cd07fd5cc317b696717acdc3c02","observation_id":"ed44b0ae-1642-48b5-bb81-d4db6d906ab3","resolution":{"observed_at":"2026-08-16T04:56:22.390432Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-16T04:56:22.398475Z","title":"The random fea ture method for solving interface problems","venue":null,"work_id":null,"year":2024},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":16,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:22.398475Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:13eca2fd6464804aac026ea072adc776d98e3fdf0508951c65027a0706668ddc","observation_id":"4fdd2e3a-d2de-40b8-a04e-5ec62e9daf2d","resolution":{"observed_at":"2026-08-16T04:56:22.398475Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-16T04:56:22.408946Z","title":null,"venue":null,"work_id":null,"year":1978},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":17,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:22.408946Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:961bf25dbe8e6fe5fc25dbe0c1545004f34da70939cba20ad17a0aa3a8ceeedd","observation_id":"5e19688b-efd1-4613-b7d9-5594ddf3e4a5","resolution":{"observed_at":"2026-08-16T04:56:22.408946Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-16T04:56:22.416939Z","title":"Learning theory: an approximation theory viewpoint , volume 24","venue":null,"work_id":null,"year":2007},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":18,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:22.416939Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:342a04a0599742e969763dc8b0045f0768ab9291996f28f334c73843b887bd90","observation_id":"7db8f9e8-95e0-47a8-8ce5-171c94da2c13","resolution":{"observed_at":"2026-08-16T04:56:22.416939Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-16T04:56:22.425991Z","title":"Approximation by superpositions of a sigmoida l function","venue":null,"work_id":null,"year":1989},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":19,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:22.425991Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:503cb30f89e8a3a37a71e0039c9d3d535c137caf960da2efddff216821d1705c","observation_id":"3465a66c-e9ea-4d2f-8037-485bb2c17127","resolution":{"observed_at":"2026-08-16T04:56:22.425991Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-16T04:56:25.897868Z","title":"Approximation theory and harmonic analysis on spheres and b alls","venue":null,"work_id":"7d60af03-8eb0-4c62-85be-cc45b0f41667","year":2013},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":20,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:22.432996Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:476d0c6f4759a8c13257a9e653ead5a58374d36eddcee0806e8bdc01f1425b88","observation_id":"30d471cc-2f91-4092-a85b-c92a340a362e","resolution":{"observed_at":"2026-08-16T04:56:25.903984Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-22T06:32:14.747728+00:00","source":"crossref"},{"observed_at":"2026-08-22T06:32:06.552537+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":"2026-08-16T04:56:25.860505Z","title":"Local randomized neural network s with hybridized discontin- uous petrov–galerkin methods for stokes–darcy ﬂows","venue":null,"work_id":"5ade5bbb-b412-409e-bfcc-6ca283175674","year":2024},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":21,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:22.451987Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:2870c14e04e0998c0d32015632d303867ecb094c61e863c69ef42ca5f33d0f4a","observation_id":"d7b08a7d-ace6-4914-9d51-b7f077e936fd","resolution":{"observed_at":"2026-08-16T04:56:25.879103Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-22T06:32:14.747728+00:00","source":"crossref"},{"observed_at":"2026-08-22T06:32:06.552537+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":"2026-08-16T04:56:25.832556Z","title":"Neural ne twork approximation","venue":null,"work_id":"03437bb4-2ea6-4ce9-8d9d-9bdcbb9fd768","year":2021},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":22,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:22.463687Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:a8978410f00f4ef36aea6167a0e671f4a0132f6033368a234d3fb4409fd8d107","observation_id":"9339ed7b-97f0-4dd2-89b2-dde402f87810","resolution":{"observed_at":"2026-08-16T04:56:25.839370Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-22T06:32:14.747728+00:00","source":"crossref"},{"observed_at":"2026-08-22T06:32:06.552537+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":"2026-08-16T04:56:25.799119Z","title":"Nonlinear approximation","venue":null,"work_id":"bcd548f9-3b5c-4cbd-b7b4-6f5fb855600c","year":1998},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":23,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:22.468564Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:52e2e0d0870a11a0e18a11fe4f51e2a81d68d39efe8549d260280ae3d968f118","observation_id":"228c7e04-d611-4458-8668-cbe33dfa084b","resolution":{"observed_at":"2026-08-16T04:56:25.806802Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-22T06:32:14.747728+00:00","source":"crossref"},{"observed_at":"2026-08-22T06:32:06.552537+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":"2026-08-16T04:56:25.772235Z","title":"Constructive approximation, volume 303","venue":null,"work_id":"ea1c5db9-b1b2-4a4e-b905-1661b40d8fa2","year":1993},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":24,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:22.476013Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:6f946d4021478ff1817f8685c59ffdd9e184c981be36ebf83f06933a531c3292","observation_id":"a22886ea-b22c-4f33-a2f2-0cb5f9106bca","resolution":{"observed_at":"2026-08-16T04:56:25.781472Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-22T06:32:14.747728+00:00","source":"crossref"},{"observed_at":"2026-08-22T06:32:06.552537+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":"2026-08-16T04:56:25.741118Z","title":"Some remarks on greed y algorithms","venue":null,"work_id":"40acb34d-d893-4fa0-97d6-3bc42ef281d6","year":1996},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":25,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:22.483200Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:2544310a9208c6cd4598a3f8d8c928aa39b214ecb70912c4c447f20be1c0c8d3","observation_id":"87f3ee52-8d37-457c-96b3-373f6edff2bf","resolution":{"observed_at":"2026-08-16T04:56:25.754674Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-22T06:32:14.747728+00:00","source":"crossref"},{"observed_at":"2026-08-22T06:32:06.552537+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":"2026-08-16T04:56:25.718682Z","title":"Local extreme learning machines and domain decomposition for solving linear and nonlinear partial diﬀerential equations","venue":null,"work_id":"c5f6f06b-ddb3-4976-95ca-7d442717dd74","year":2021},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":26,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:22.489800Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:e7ade94d1278026e2b63a2a99df3e3e95c816138827a67e5ec9dbde14e67ef34","observation_id":"59d74ec3-c34f-426b-8876-39eec655924e","resolution":{"observed_at":"2026-08-16T04:56:25.724690Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-22T06:32:14.747728+00:00","source":"crossref"},{"observed_at":"2026-08-22T06:32:06.552537+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":"2026-08-16T04:56:25.699890Z","title":"Physics informed extreme le arning machine (pielm)–a rapid method for the numerical solution of partial diﬀerential equa tions","venue":null,"work_id":"b66a601e-2001-45b0-a9b2-ad4066332c24","year":2020},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":27,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:22.495412Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:d02038cbc588dd129305320e1a8d3446f62898fb7da560c14a5598c5352f0bf5","observation_id":"d994280d-4fbb-4ebd-8547-711705fc4fde","resolution":{"observed_at":"2026-08-16T04:56:25.706642Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-22T06:32:14.747728+00:00","source":"crossref"},{"observed_at":"2026-08-22T06:32:06.552537+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":"2026-08-16T04:56:25.666101Z","title":"A priori estimates of the populat ion risk for two-layer neural networks","venue":null,"work_id":"524796bb-ed85-4702-8760-4344dddc0aba","year":2019},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":28,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:22.510674Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:4d869db2c9fd1c5d8c67d49f2b445d5eab3be3edd013fed934d35c384fbf515f","observation_id":"7161625c-2e95-4f6a-8661-608cab7c5ea2","resolution":{"observed_at":"2026-08-16T04:56:25.677282Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-22T06:32:14.747728+00:00","source":"crossref"},{"observed_at":"2026-08-22T06:32:06.552537+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":"2026-08-16T04:56:25.626601Z","title":"The barron space and the ﬂow-in duced function spaces for neural network models","venue":null,"work_id":"144d5c2d-a0aa-415e-adc0-479bbbfcd23b","year":2022},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":29,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:22.527245Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:d519c9a377608c64b938fc07fd91448ae12f36c82aa4087d2d22793cea62e588","observation_id":"e72c7e82-bef5-4de8-af79-1b46f694261d","resolution":{"observed_at":"2026-08-16T04:56:25.640784Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-22T06:32:14.747728+00:00","source":"crossref"},{"observed_at":"2026-08-22T06:32:06.552537+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":"2026-08-16T04:56:25.575209Z","title":"Representation formulas and pointwise properties for barron functions","venue":null,"work_id":"28b1f83b-5a9e-4a68-9a25-0fbcf8c0ce92","year":2022},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":30,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:22.536667Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:4de0a01b75ef83e7008a5a1a633b6a285961076b5a200dcd30c23ecd776c41f8","observation_id":"3afd9ce4-73c1-41ab-a73d-849647eb3ff4","resolution":{"observed_at":"2026-08-16T04:56:25.584671Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-22T06:32:14.747728+00:00","source":"crossref"},{"observed_at":"2026-08-22T06:32:06.552537+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":"2026-08-16T04:56:25.538462Z","title":"Generalisation error in learning with random features and the hidden manifold model","venue":null,"work_id":"5e4c8ede-548f-4eb6-8e79-173b9660ffc3","year":2020},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":31,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:22.552526Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:8fdb318b83377d06af527ee302ca3055a533323d5f219f45624062bd03d94f08","observation_id":"13bce886-e5ab-426b-b875-72df9ba35e2b","resolution":{"observed_at":"2026-08-16T04:56:25.547727Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-22T06:32:14.747728+00:00","source":"crossref"},{"observed_at":"2026-08-22T06:32:06.552537+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":"2026-08-16T04:56:25.507852Z","title":"Deep neural networks with random gaussian weights: A universal classiﬁcation strategy? IEEE Transactions on Signal Process- ing, 64(13):3444–3457, 2016","venue":null,"work_id":"5c7e78e8-4362-45eb-b974-1a0ec45b5e14","year":2016},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":32,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:22.560819Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:bfd3542ef76e30c08c53b6ebca8df5ad4260ff3f77f3edcdcadccdb0f9c930bf","observation_id":"e9d73138-b85b-4db5-8d32-71fe64e4be60","resolution":{"observed_at":"2026-08-16T04:56:25.514918Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-22T06:32:14.747728+00:00","source":"crossref"},{"observed_at":"2026-08-22T06:32:06.552537+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":"2026-08-16T04:56:25.481579Z","title":"Delving de ep into rectiﬁers: Surpassing human-level performance on imagenet classiﬁcation","venue":null,"work_id":"566038d3-f75d-40ef-b805-9ea7d54a08ed","year":2015},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":33,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:22.575836Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:083d4d6ad0da2d7549e434c002b9fd2fd8c58eb1936a0bd045eba90b473279ea","observation_id":"fa44148c-e92c-4aa9-9eec-cb1b835d61f4","resolution":{"observed_at":"2026-08-16T04:56:25.489773Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-22T06:32:14.747728+00:00","source":"crossref"},{"observed_at":"2026-08-22T06:32:06.552537+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-16T04:56:22.591301Z","title":"Multilayer feedforward networks are universal approximators","venue":null,"work_id":null,"year":1989},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":34,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:22.591301Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:6376118a8a99f4841f9357740025239cc9d59adb2dd7c348cb03f1dd63ae30bd","observation_id":"c6f7fa84-9f96-41af-b4ca-a7ce32ffebe0","resolution":{"observed_at":"2026-08-16T04:56:22.591301Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-16T04:56:25.419258Z","title":"Universality laws for high-dimensional learn ing with random features","venue":null,"work_id":"425962b7-f695-46a7-b262-c53ca6c954cd","year":1932},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":35,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:22.598162Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:5b12f0ec17f0057f8069176e48a714af4d62ca417639c8ab2ac07cb70165e7a5","observation_id":"0c39e3de-79c5-4ddc-a6ca-788fce4e6c26","resolution":{"observed_at":"2026-08-16T04:56:25.430208Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-22T06:32:14.747728+00:00","source":"crossref"},{"observed_at":"2026-08-22T06:32:06.552537+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":"2026-08-16T04:56:25.365716Z","title":"Universal approximation using incre- mental constructive feedforward networks with random hidden n odes","venue":null,"work_id":"a6a71f60-d8c3-4f64-890f-d453e4474514","year":2006},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":36,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:22.607741Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:3262c5a8f369491c43360e0a56b41c27e2b1352a30dc4b20853bdd5f34cc5d30","observation_id":"70b7a7fa-5e6b-4ff6-8c66-0b4161aeabea","resolution":{"observed_at":"2026-08-16T04:56:25.378098Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-22T06:32:14.747728+00:00","source":"crossref"},{"observed_at":"2026-08-22T06:32:06.552537+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":"2026-08-16T04:56:25.336030Z","title":"Extreme learning machine: theory and applications","venue":null,"work_id":"6cadfe20-ed72-4a02-aaf1-d008a5b52f88","year":2006},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":37,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:22.623074Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:eb353930435ccf7143a799da5254e66d69552515222918c66c391e6329e85381","observation_id":"2818f8e8-4553-4ca7-8fa0-cbc31a0ea88c","resolution":{"observed_at":"2026-08-16T04:56:25.347468Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-22T06:32:14.747728+00:00","source":"crossref"},{"observed_at":"2026-08-22T06:32:06.552537+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":"2026-08-16T04:56:25.309462Z","title":"Stochastic choice of basis funct ions in adaptive function ap- proximation and the functional-link net","venue":null,"work_id":"939a07e5-fc3e-4020-8c27-a9cf8067b598","year":1995},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":38,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:22.630048Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:47b5e87f175c13eec70d3138b570a8c4c496af3fdb182d0e9c237bf9bd68bc48","observation_id":"a0dc9ed8-f408-4ed1-a26a-c3bc773efca5","resolution":{"observed_at":"2026-08-16T04:56:25.315700Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-22T06:32:14.747728+00:00","source":"crossref"},{"observed_at":"2026-08-22T06:32:06.552537+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":"2026-08-16T04:56:25.276225Z","title":"Norming sets and spherica l cubature formulas","venue":null,"work_id":"86bb2ced-5bdb-4710-8bfb-42acacecb778","year":2023},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":39,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:22.645054Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:c0057c02e3f83d139e25c71794c6bf1a16c7927e5d4240c53454e1fab5cdbee4","observation_id":"536914fb-ee47-4f63-a3a7-0ee8fa761ff3","resolution":{"observed_at":"2026-08-16T04:56:25.283409Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-22T06:32:14.747728+00:00","source":"crossref"},{"observed_at":"2026-08-22T06:32:06.552537+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":"2026-08-16T04:56:25.250339Z","title":"A simple lemma on greedy approximation in hilbert spac e and convergence rates for projection pursuit regression and neural network tra ining","venue":null,"work_id":"d3d1230c-46a9-4610-aae7-d877890fdbcf","year":1992},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":40,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:22.653781Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:e7db1bc699f6ffd6f2b31f039b5b1559379d4fc5628c6b921a625f560689088c","observation_id":"9df33da2-1817-4d73-b1ca-abaf4036dec9","resolution":{"observed_at":"2026-08-16T04:56:25.257264Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-22T06:32:14.747728+00:00","source":"crossref"},{"observed_at":"2026-08-22T06:32:06.552537+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":"2026-08-16T04:56:25.216720Z","title":"Approximation by comb inations of relu and squared relu ridge functions with l1 and l0 controls","venue":null,"work_id":"808687ed-b1c4-43aa-8e2d-4fa98bc835d3","year":2018},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":41,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:22.659336Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:6717a70d602e57acc9b36d34748e6767d5dfe02518b7774f2a4dfa197a0d6155","observation_id":"747c5a75-06b4-438d-a956-db01de308d58","resolution":{"observed_at":"2026-08-16T04:56:25.225839Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-22T06:32:14.747728+00:00","source":"crossref"},{"observed_at":"2026-08-22T06:32:06.552537+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":"2026-08-16T04:56:25.181572Z","title":"On linear dimensionality of topolog ical vector spaces","venue":null,"work_id":"b1d35bab-9bbe-4a4e-8084-470d7de15ca3","year":null},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":42,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:22.677249Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:b5f422c9aefcc489e637ffccbebaadd206f4d92c535dcf6c54a3361f1dedbc25","observation_id":"0bfb0bcf-9f30-45f2-8547-ae379bd53934","resolution":{"observed_at":"2026-08-16T04:56:25.188837Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-22T06:32:14.747728+00:00","source":"crossref"},{"observed_at":"2026-08-22T06:32:06.552537+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":"2026-08-16T04:56:25.143048Z","title":"Some problems in the theory of ridge functions","venue":null,"work_id":"15bd07c8-01d0-4b59-993c-07892ee28587","year":2018},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":43,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:22.685971Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:ac15d4a919bde5f3ac365705b894bef1cf525442a53839cbe555b95b747b00ae","observation_id":"20173c8f-b720-4775-851e-24ed8bac5243","resolution":{"observed_at":"2026-08-16T04:56:25.154226Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-22T06:32:14.747728+00:00","source":"crossref"},{"observed_at":"2026-08-22T06:32:06.552537+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":"2026-08-16T04:56:25.115007Z","title":"K˚ urkov´ a and M","venue":null,"work_id":"fc358022-62fd-4df6-afa1-acaaa69b7df6","year":2001},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":44,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:22.693599Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:fd0cf67940648e9144cb85388e41ff430b4b0543e18564d64d47d4fd22ae44f7","observation_id":"04aa3a7f-b622-4704-b7e5-7e844f61f6e4","resolution":{"observed_at":"2026-08-16T04:56:25.121262Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-22T06:32:14.747728+00:00","source":"crossref"},{"observed_at":"2026-08-22T06:32:06.552537+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":"2026-08-16T04:56:25.085966Z","title":"K˚ urkov´ a and M","venue":null,"work_id":"33a7e256-161c-43fa-a0ea-cd771929690f","year":2002},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":45,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:22.705265Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:940a48837353399d62cb41a4a2edf37c7ed692fbace52e4f1b47c371bad5d676","observation_id":"2f1d50eb-fbd0-4bd2-90cd-a9e99514fe3d","resolution":{"observed_at":"2026-08-16T04:56:25.093004Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-22T06:32:14.747728+00:00","source":"crossref"},{"observed_at":"2026-08-22T06:32:06.552537+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":"2026-08-16T04:56:25.051552Z","title":"Deep learning","venue":null,"work_id":"e2f6719a-de93-4e3b-9323-88ed981bf0d6","year":2015},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":46,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:22.713411Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:43d6d74843ef709175dcdaf520cc898a498283d67a5ae1660144ed16c55892ef","observation_id":"23cca422-a4ce-4698-ab6b-1911d59fe88e","resolution":{"observed_at":"2026-08-16T04:56:25.060811Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-22T06:32:14.747728+00:00","source":"crossref"},{"observed_at":"2026-08-22T06:32:06.552537+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":"2026-08-16T04:56:25.012779Z","title":"Multilayer feedforward networks with a nonpolynomial activation function can approximate any function","venue":null,"work_id":"e8b15a4e-f11c-4b91-92f4-bca76e875c8a","year":1993},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":47,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:22.719612Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:f96891a94bfa8aa7fd69a8c66696498d3800c5b3376311640e071e949c5417f9","observation_id":"489deaae-ed07-4cc7-82bc-04fdcd62d75d","resolution":{"observed_at":"2026-08-16T04:56:25.020625Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-22T06:32:14.747728+00:00","source":"crossref"},{"observed_at":"2026-08-22T06:32:06.552537+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":"2026-08-16T04:56:24.983977Z","title":"Approximation of funct ions of ﬁnite variation by superpositions of a sigmoidal function","venue":null,"work_id":"ac23072d-3e36-43c1-b5f5-bd73c6ac4318","year":2004},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":48,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:22.739097Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:d7d243ccb1053c449b73224d7aad6deee330d369d756e9a5799ed3af89fc48b7","observation_id":"b3523eae-c7e7-41d6-aa94-51f1675ebe59","resolution":{"observed_at":"2026-08-16T04:56:24.994875Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-22T06:32:14.747728+00:00","source":"crossref"},{"observed_at":"2026-08-22T06:32:06.552537+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":"2026-08-16T04:56:24.954625Z","title":"Towar ds a uniﬁed analysis of random fourier features","venue":null,"work_id":"b154448a-d6aa-443c-9b1d-e195ea582d9a","year":2019},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":49,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:22.747138Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:96ca315a139a30a6af23ffeb3f8cdf745cd593f14fe587d95a4d39e569389372","observation_id":"80299d00-6e96-4ac6-86d9-b9640aeefe89","resolution":{"observed_at":"2026-08-16T04:56:24.964732Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-22T06:32:14.747728+00:00","source":"crossref"},{"observed_at":"2026-08-22T06:32:06.552537+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":"2026-08-16T04:56:24.922461Z","title":"Lower bounds of the discretiz ation error for piecewise polynomials","venue":null,"work_id":"3ec6d8d8-9b4d-486a-aa20-9be599090ad3","year":2014},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":50,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:22.755456Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:76ef4febc366150059675a4efe476bdc530a8c6222d0481e9d72dd90bbf25adf","observation_id":"03a8b301-6132-43a6-bc50-25c2fb69af77","resolution":{"observed_at":"2026-08-16T04:56:24.933062Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-22T06:32:14.747728+00:00","source":"crossref"},{"observed_at":"2026-08-22T06:32:06.552537+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":"2026-08-16T04:56:24.894775Z","title":"Is extreme lea rning machine feasible? a theoretical assessment (part 1)","venue":null,"work_id":"d9c79fc1-72dc-40d3-bb70-2e790ff44adc","year":2014},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":51,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:22.762628Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:928ac36b7feea2428d7f684fe450b9c2e4d93eb1323ae0f393af8d7b87d86f13","observation_id":"f835c6c1-723f-41da-96f2-f4f7cb71ff4c","resolution":{"observed_at":"2026-08-16T04:56:24.903109Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-22T06:32:14.747728+00:00","source":"crossref"},{"observed_at":"2026-08-22T06:32:06.552537+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":"2026-08-16T04:56:24.871621Z","title":"Randomized nonlinear component analysis","venue":null,"work_id":"4d8ac661-eb5e-495e-a202-3220a5582a95","year":2014},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":52,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:22.770621Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:df3dc694dded5430088bc528d559e2790f216aeb567a6b36bd0a3c62a20494d3","observation_id":"5777a9dc-6691-4f53-a504-65d3666f256c","resolution":{"observed_at":"2026-08-16T04:56:24.878306Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-22T06:32:14.747728+00:00","source":"crossref"},{"observed_at":"2026-08-22T06:32:06.552537+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":"2026-08-16T04:56:24.839705Z","title":"Dee p neural networks with ﬁxed width can be universal approximators","venue":null,"work_id":"d9c63c1f-155c-4c17-abd6-57b2f3559178","year":2021},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":53,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:22.776922Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:62cb8f094e9994ec5eeeba3a3d9b45fb2d5b6f8eb306f1ca49d2a83de36826ae","observation_id":"6661cc91-f481-464b-986e-ce1c4a3b22a1","resolution":{"observed_at":"2026-08-16T04:56:24.847821Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-22T06:32:14.747728+00:00","source":"crossref"},{"observed_at":"2026-08-22T06:32:06.552537+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":"2026-08-16T04:56:24.812429Z","title":"Uniform approximatio n rates and metric entropy of shallow neural networks","venue":null,"work_id":"884ae48c-e833-43f2-9cb5-965c8812f476","year":2022},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":54,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:22.782990Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:65585f06d9ec354820fe166e6753ad3528dbd622c68afbdaa75e3adbd14c2fc0","observation_id":"29748f2e-8ec2-4b2a-8cd3-c8e08aea4d0f","resolution":{"observed_at":"2026-08-16T04:56:24.819703Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-22T06:32:14.747728+00:00","source":"crossref"},{"observed_at":"2026-08-22T06:32:06.552537+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":"2026-08-16T04:56:24.782696Z","title":"On the near optimality of the stocha stic approximation of smooth functions by neural networks","venue":null,"work_id":"db777e68-4e3a-4088-b188-b1b96f20e7ff","year":2000},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":55,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:22.788882Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:bc401ff6a144dc43ee60575f385e37acd95593e44a65d41d2ed96ee58cb81dcc","observation_id":"42547d2a-c978-4ec4-bbdf-3d1e7904b156","resolution":{"observed_at":"2026-08-16T04:56:24.791236Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-22T06:32:14.747728+00:00","source":"crossref"},{"observed_at":"2026-08-22T06:32:06.552537+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":"2026-08-16T04:56:24.755909Z","title":"Random approximants and neural networks","venue":null,"work_id":"d65c875b-b2f9-484b-873d-08d3e4c8e19d","year":1996},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":56,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:22.801308Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:522376f383aaeefd45a462ac83eb735c7bec3ee0dc58fee2b82fd99e3986fcef","observation_id":"b335675d-42dd-4812-9667-966a74bce09a","resolution":{"observed_at":"2026-08-16T04:56:24.763543Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-22T06:32:14.747728+00:00","source":"crossref"},{"observed_at":"2026-08-22T06:32:06.552537+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":"2026-08-16T04:56:24.732983Z","title":"Uniform approximation by neural networks","venue":null,"work_id":"e37f93d5-08fe-42af-877e-fdc3c5ffede3","year":1998},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":57,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:22.811059Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:9b67c503c10e3d8d872f27a0894e240ff35986dddc7dbc90282b3b23f5239294","observation_id":"307fc214-c595-450d-8f91-cfbb46a80b59","resolution":{"observed_at":"2026-08-16T04:56:24.740063Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-22T06:32:14.747728+00:00","source":"crossref"},{"observed_at":"2026-08-22T06:32:06.552537+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-16T04:56:22.816924Z","title":"Approximation rat es for shallow reluk neural networks on sobolev spaces via the radon transform","venue":null,"work_id":null,"year":2024},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":58,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:22.816924Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:9ce88f4a77404e0e2bedf1da8b8760696cd38843cbff920073b295d809be6663","observation_id":"914ce906-dc54-4bb9-b810-f5a536fd036e","resolution":{"observed_at":"2026-08-16T04:56:22.816924Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-16T04:56:24.711195Z","title":"Do neural networks have better app roximation properties than polynomials or ﬁnite elements for high-dimensional problems? preprint, 2025","venue":null,"work_id":"55a68932-59a7-45c1-9ea7-7aae7da380c6","year":2025},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":59,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:22.824945Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:1dcea7ca4591996ead1ef195abff7109fcdbd840d5a77b8d028fd1ca8afd1b2d","observation_id":"111a5bfd-8b68-4137-a485-eab1cab615d1","resolution":{"observed_at":"2026-08-16T04:56:24.718731Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-22T06:32:14.747728+00:00","source":"crossref"},{"observed_at":"2026-08-22T06:32:06.552537+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":"2026-08-16T04:56:24.680726Z","title":"Rates of approximation by relu sh allow neural networks","venue":null,"work_id":"0cb35309-100c-4d34-b452-d482b06485fd","year":2023},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":60,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:22.832524Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:43966a10d59c14cbb64f8d33dadb30ae69ca6670022f84d3af6b20db676ddad4","observation_id":"db29cbca-3d22-4a3c-87d5-8c46fd560acc","resolution":{"observed_at":"2026-08-16T04:56:24.689733Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-22T06:32:14.747728+00:00","source":"crossref"},{"observed_at":"2026-08-22T06:32:06.552537+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":"2026-08-16T04:56:24.645872Z","title":"Type et cotype dans les espaces munis de structure s locales inconditionnelles","venue":null,"work_id":"1d74ce51-b289-4ace-a2cf-47df6fe9243b","year":1973},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":61,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:22.845618Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:a112171a6bb0bacb900923e3cc44686529c5af312b78c25fc9ced6caec3583ef","observation_id":"6875ea4a-8537-4e76-84f2-9da399e61e72","resolution":{"observed_at":"2026-08-16T04:56:24.655444Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-22T06:32:14.747728+00:00","source":"crossref"},{"observed_at":"2026-08-22T06:32:06.552537+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":"2026-08-16T04:56:24.611279Z","title":"The generalization error of ra ndom features regression: Precise asymptotics and the double descent curve","venue":null,"work_id":"5188c0b1-5fa7-49ef-82c2-e69801a8bc99","year":2022},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":62,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:22.853738Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:986378d74ffef71ce1cfdf9b68d9625bc03a59141c26c1c51fdd9676d0100a79","observation_id":"e9e8239e-3c36-40bb-b7ba-7cc6f0d06c72","resolution":{"observed_at":"2026-08-16T04:56:24.619847Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-22T06:32:14.747728+00:00","source":"crossref"},{"observed_at":"2026-08-22T06:32:06.552537+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":"2026-08-16T04:56:24.576450Z","title":"A new function space from barron cla ss and application to neural network approximation","venue":null,"work_id":"462ffb53-3b12-4522-8bcb-2af9b0c25f8e","year":2022},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":63,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:22.863083Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:03b02eecf37995185179e95a123be4caa9934c5dd9ddae364c5cfda94c3905db","observation_id":"7b2322eb-402a-49e5-9f32-a1786ff0cf8e","resolution":{"observed_at":"2026-08-16T04:56:24.589618Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-22T06:32:14.747728+00:00","source":"crossref"},{"observed_at":"2026-08-22T06:32:06.552537+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":"2026-08-16T04:56:24.537239Z","title":"Spherical marcinkiewicz-z ygmund inequalities and positive quadrature","venue":null,"work_id":"b1addaff-4615-4938-b1ac-5446c94db8dc","year":2001},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":64,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:22.869744Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:b38bf4d238d5934b337214ba07669e36ba5f7357142a41afe7f5f9030aa2a2a1","observation_id":"c828ed18-9406-46d6-8d89-6759850ab50c","resolution":{"observed_at":"2026-08-16T04:56:24.553757Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-22T06:32:14.747728+00:00","source":"crossref"},{"observed_at":"2026-08-22T06:32:06.552537+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":{"arxiv_id":"2308.03230","last_updated":"2023-12-10T19:07:35Z","snapshot_observed_at":"2026-08-17T14:53:02.780809Z","submitted_at":"2023-08-07T00:14:46Z","title":"Tractability of approximation by general shallow networks","version":2},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2308.03230","snapshot_observed_at":"2026-08-16T04:56:22.877068Z","title":"Tractability of approximatio n by general shallow net- works","venue":null,"work_id":null,"year":2023},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":65,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:22.877068Z"},"links":{"cited_paper":"/paper/2308.03230","citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:81b12b15d8003631f1fcce9b6408ac3497fe1a7abd0c6814bbff28721cf98e71","observation_id":"faa8a98d-ca0d-48c7-99f9-1a45f167ff6f","resolution":{"observed_at":"2026-08-16T04:56:22.877068Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-16T04:56:24.499613Z","title":"Eignets for function approximation on m anifolds","venue":null,"work_id":"27c6a0d2-44a8-4b55-a483-19d3f40b9faf","year":2010},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":66,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:22.885584Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:1b90e247e65fd8b40eeec274a9b6c3ad9d18ce979c59a9b33bc1046f1727f3d4","observation_id":"5ea16f13-be0a-4a1d-9fdc-9aa16e14c947","resolution":{"observed_at":"2026-08-16T04:56:24.512286Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-22T06:32:14.747728+00:00","source":"crossref"},{"observed_at":"2026-08-22T06:32:06.552537+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":"2026-08-16T04:56:24.467757Z","title":"Kernel-based analysis of massive data","venue":null,"work_id":"e48473cb-f002-489c-8de7-79f92dbf6cf1","year":2020},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":67,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:22.892658Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:dd28bf6f241d732387efdeaa95a6a6490f1de3ad5caf3f3ad6cc6131bdfc24bf","observation_id":"95b37b54-d98f-4790-be5a-25242e3fe12d","resolution":{"observed_at":"2026-08-16T04:56:24.480495Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-22T06:32:14.747728+00:00","source":"crossref"},{"observed_at":"2026-08-22T06:32:06.552537+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":"2026-08-16T04:56:24.433037Z","title":"Approximation properties of a mu ltilayered feedforward artiﬁcial neural network","venue":null,"work_id":"7218117e-233d-4311-9817-011f47500eac","year":1993},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":68,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:22.899809Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:bb9adf2e57c307451be0d42ea44f4d80965ce8e85a6e2e3e340042c915200b98","observation_id":"17ca3ea6-3d54-4e42-80ce-3b01dcb364c6","resolution":{"observed_at":"2026-08-16T04:56:24.441117Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-22T06:32:14.747728+00:00","source":"crossref"},{"observed_at":"2026-08-22T06:32:06.552537+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":"2026-08-16T04:56:24.405639Z","title":"Weighted quadrature formulas a nd approximation by zonal function networks on the sphere","venue":null,"work_id":"fb60107b-2e2b-4db7-96dd-dfbe806adb8a","year":2006},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":69,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:22.906458Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:94eb6d9596e6f59431ed1da6f7c2449d6985fb40ca8fe17ec9d6c676f2dfd1f8","observation_id":"86fc4fef-ab3d-40ef-83d1-0f325849ac9d","resolution":{"observed_at":"2026-08-16T04:56:24.416187Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-22T06:32:14.747728+00:00","source":"crossref"},{"observed_at":"2026-08-22T06:32:06.552537+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":"2026-08-16T04:56:24.365650Z","title":"Foundations of Machine Learn- ing","venue":null,"work_id":"b30285a5-c676-4ff6-b299-385cea0b383d","year":2018},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":70,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:22.912453Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:c15e82f75a10f081bf2af4c0348f94752399cdc8847fafb2c7b97efd7f5573e5","observation_id":"d19373c7-f979-4d32-bf0f-765373266656","resolution":{"observed_at":"2026-08-16T04:56:24.380350Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-22T06:32:14.747728+00:00","source":"crossref"},{"observed_at":"2026-08-22T06:32:06.552537+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":"2026-08-16T04:56:24.336835Z","title":"The random feature mod el for input-output maps between banach spaces","venue":null,"work_id":"4714e4d5-65a2-4e77-b300-190e11d8526f","year":2021},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":71,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:22.920572Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:1edc498d01eead86979a2f22e9e08d55b1ce90774f133976f8a9ccbf9237c893","observation_id":"ee6c6d9f-9968-4ff5-9829-51654cbf90f2","resolution":{"observed_at":"2026-08-16T04:56:24.342707Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-22T06:32:14.747728+00:00","source":"crossref"},{"observed_at":"2026-08-22T06:32:06.552537+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":"2026-08-16T04:56:24.312661Z","title":"Learnin g and generalization charac- teristics of the random vector functional-link net","venue":null,"work_id":"106254e6-8c75-431c-928b-709888cfbf31","year":1994},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":72,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:22.927462Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:cad3bcd05c5eb809d31afe1064d9c348755cf5b5de409c57d8fde9f31353f992","observation_id":"1163a465-81e5-4c3a-816e-01010a938f28","resolution":{"observed_at":"2026-08-16T04:56:24.320065Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-22T06:32:14.747728+00:00","source":"crossref"},{"observed_at":"2026-08-22T06:32:06.552537+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":"2026-08-16T04:56:24.268612Z","title":"Approximation by ridge functions and neu ral networks","venue":null,"work_id":"fd12aa44-f8c5-4d8c-8a32-31ab2c548101","year":1998},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":73,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:22.934766Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:2f6283a514254dd10162e7628bef5558aa19751b138467dd4be4a610eb0d8fc4","observation_id":"88f60b10-2816-474d-b259-ecb8e6e2ea14","resolution":{"observed_at":"2026-08-16T04:56:24.274735Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-22T06:32:14.747728+00:00","source":"crossref"},{"observed_at":"2026-08-22T06:32:06.552537+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":"2026-08-16T04:56:24.237387Z","title":"Approximation theory of the mlp model in neural net works","venue":null,"work_id":"3af280e1-c8b1-4ffc-8f06-ec0b1f449a5c","year":1999},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":74,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:22.943955Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:cb963ff7186cbe231651563c2709a898e04eb0a365eedcde37bd9f28210108e6","observation_id":"cf41abd8-78bd-4f68-88d2-22c5e0d8f23c","resolution":{"observed_at":"2026-08-16T04:56:24.246818Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-22T06:32:14.747728+00:00","source":"crossref"},{"observed_at":"2026-08-22T06:32:06.552537+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":"2026-08-16T04:56:24.183370Z","title":"Remarques sur un r´ esultat non publi´ e de B","venue":null,"work_id":"2e3fc5a5-0f02-4a3e-9878-df74112279a1","year":1981},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":75,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:22.950832Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:2f8f99dda8d608ea538ea8a48e748e6c087da9bdaf13e74c2592fa0f274ef7d5","observation_id":"170c1e0e-78c9-409b-b793-601106c13164","resolution":{"observed_at":"2026-08-16T04:56:24.193960Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-22T06:32:14.747728+00:00","source":"crossref"},{"observed_at":"2026-08-22T06:32:06.552537+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":"2026-08-16T04:56:24.152177Z","title":"Random features for large-sca le kernel machines","venue":null,"work_id":"535e93b1-7f27-4b13-892c-f52b0491029b","year":2007},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":76,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:22.958339Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:fb2a6e50e012910b65233d3f2710c630731336e5f5fffcc867be37be5293d68b","observation_id":"f66d4585-efe3-42f5-84bf-19c5e1aa8631","resolution":{"observed_at":"2026-08-16T04:56:24.161524Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-22T06:32:14.747728+00:00","source":"crossref"},{"observed_at":"2026-08-22T06:32:06.552537+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":"2026-08-16T04:56:24.118343Z","title":"Uniform approximation of functio ns with random bases","venue":null,"work_id":"138fee9a-6426-4cc2-9f69-f89790d99913","year":2008},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":77,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:22.970421Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:f43aab9213b3cc6e86805026914d30ea6c8a33cea02ab437a2b18fd25bf03a4d","observation_id":"52822f5b-9479-48d2-b5bc-63c18188186e","resolution":{"observed_at":"2026-08-16T04:56:24.126877Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-22T06:32:14.747728+00:00","source":"crossref"},{"observed_at":"2026-08-22T06:32:06.552537+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":"2026-08-16T04:56:24.091720Z","title":"Weighted sums of random kitchen sinks: Replacing mini- mization with randomization in learning","venue":null,"work_id":"deff18f6-4f74-46cd-be3c-06d289fc2ec6","year":2008},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":78,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:22.980176Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:57db173e5676c9eaecf7831962fe60691dd93d091df4637a8d2cbb46fd84d16f","observation_id":"8314a470-30e1-4d43-9672-d8b91e082495","resolution":{"observed_at":"2026-08-16T04:56:24.098332Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-22T06:32:14.747728+00:00","source":"crossref"},{"observed_at":"2026-08-22T06:32:06.552537+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":"2026-08-16T04:56:24.064123Z","title":"On random weights and unsupervised feature learning","venue":null,"work_id":"bedd3e6f-962b-49d7-9a5c-de6e6cf6132b","year":2011},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":79,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:22.989148Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:3025da39c76bd084fc6442261bd985c462fe58de978568337ac0f943a2dfba1d","observation_id":"8d622db9-c0eb-4ce8-85a3-d84d249ad9b9","resolution":{"observed_at":"2026-08-16T04:56:24.073584Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-22T06:32:14.747728+00:00","source":"crossref"},{"observed_at":"2026-08-22T06:32:06.552537+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":"2026-08-16T04:56:24.030433Z","title":"Zu einem problem von shephard ¨ uber die projek tionen konvexer k¨ orper","venue":null,"work_id":"ad6a1059-bf24-4c27-910a-5cf99aa9ec87","year":1967},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":80,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:22.996999Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:d3252b50ffcd529e5a493e2efaa5e475d6c8e9d7e81985fa914515bd4a36ebdd","observation_id":"1c865c19-8715-47f5-8a10-952dbe9118a6","resolution":{"observed_at":"2026-08-16T04:56:24.044685Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-22T06:32:14.747728+00:00","source":"crossref"},{"observed_at":"2026-08-22T06:32:06.552537+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-16T04:56:23.007044Z","title":"Understanding machine learning: From theory to algorithms","venue":null,"work_id":null,"year":2014},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":81,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:23.007044Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:72de40160dc9d1cab9a0be79ce941eac9aaff43b416a2ee99fda568270bb03ae","observation_id":"2c1167d2-34f8-480c-837f-b230219135dc","resolution":{"observed_at":"2026-08-16T04:56:23.007044Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-16T04:56:23.954077Z","title":"Optimal approximation rates for deep relu ne ural networks on sobolev and besov spaces","venue":null,"work_id":"826feb71-22d4-4fd6-8f4e-5358bf7f8abb","year":2023},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":82,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:23.014148Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:416a42663b5859dbdabe0cf8154ee7026e0c21f2d409b12000690ed9a9043658","observation_id":"ab23d029-dada-4e79-91ed-f80ab012ff13","resolution":{"observed_at":"2026-08-16T04:56:23.964851Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-22T06:32:14.747728+00:00","source":"crossref"},{"observed_at":"2026-08-22T06:32:06.552537+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":"2026-08-16T04:56:23.921893Z","title":"Greedy training algorithms for neural networks and applications to pdes","venue":null,"work_id":"2fbc57ab-22c6-49ec-929e-ec7e4012eb0a","year":2023},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":84,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:23.029119Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:9686b481db56d50f99ba555deb9000890cb5c00d3b4184d0118c4cd12aedf77b","observation_id":"161257ff-e4ba-46ad-9013-97beb1c861c1","resolution":{"observed_at":"2026-08-16T04:56:23.932305Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-22T06:32:14.747728+00:00","source":"crossref"},{"observed_at":"2026-08-22T06:32:06.552537+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":"2026-08-16T04:56:23.890969Z","title":"High-order approximation ra tes for shallow neural net- works with cosine and ReLUk activation functions","venue":null,"work_id":"f6cf63d1-4462-4feb-a18e-7caad2fe0c23","year":2022},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":85,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:23.036693Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:60bf224a087903e9172b5b7471d376f4e9f38dc048e4ad3742801d47af10d698","observation_id":"6aab6b51-5d71-4e4d-9eb3-e2f42a80b11a","resolution":{"observed_at":"2026-08-16T04:56:23.899426Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-22T06:32:14.747728+00:00","source":"crossref"},{"observed_at":"2026-08-22T06:32:06.552537+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":"2026-08-16T04:56:23.863704Z","title":"Optimal convergence rates for the orthogonal greedy algorithm","venue":null,"work_id":"f5a5f1f8-1f3e-4677-8860-4639f0404d57","year":2022},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":86,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:23.046338Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:29ba0ec04ac2e9c91bba83fae5aa3ee7740728ede37dcbe65b7aeca2951b7e65","observation_id":"fb2c3c09-8e1f-4966-8637-466c6d8a8733","resolution":{"observed_at":"2026-08-16T04:56:23.873238Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-22T06:32:14.747728+00:00","source":"crossref"},{"observed_at":"2026-08-22T06:32:06.552537+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":"2026-08-16T04:56:23.825754Z","title":"Sharp bounds on the approx imation rates, metric entropy, and n-widths of shallow neural networks","venue":null,"work_id":"508f0e0d-b494-4598-bffc-c809d5c210ad","year":2022},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":87,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:23.056635Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:8a377626fd95eec72749ee4f23dab38683ee58ebd01c5652b21c7f3c6320ca61","observation_id":"59cde62c-20c5-40cc-8865-76343c7634d0","resolution":{"observed_at":"2026-08-16T04:56:23.835267Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-22T06:32:14.747728+00:00","source":"crossref"},{"observed_at":"2026-08-22T06:32:06.552537+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":"2026-08-16T04:56:23.786529Z","title":"Characterization of the var iation spaces corresponding to shallow neural networks","venue":null,"work_id":"f0965dcc-63c7-419d-a14f-55377ce6f98d","year":2023},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":88,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:23.066641Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:beccf1e24ba8096461104053fe1858bc3ae34f403d6b962b9b515d5dee75e168","observation_id":"265f6031-911f-4856-bac9-d789ac456570","resolution":{"observed_at":"2026-08-16T04:56:23.795580Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-22T06:32:14.747728+00:00","source":"crossref"},{"observed_at":"2026-08-22T06:32:06.552537+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-16T04:56:23.072846Z","title":"Singular integrals and diﬀerentiability properties of fun ctions","venue":null,"work_id":null,"year":1970},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":89,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:23.072846Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:086c246284ceb5be9e15905c289846e08a87b7a75be6ed6f9550e49a0cc79698","observation_id":"2b6a0765-e825-4698-91b5-c9872b6a5b72","resolution":{"observed_at":"2026-08-16T04:56:23.072846Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-16T04:56:23.744648Z","title":"Introduction to Fourier analysis on Euclidean spaces , vol- ume 1","venue":null,"work_id":"49b7d9fe-413e-4e92-9c3c-fb26d62544e9","year":1971},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":90,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:23.078424Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:14a776285dfe5f6393e4e26a3a5470ed3f578c51a1001a48972e4b49cd23ec59","observation_id":"cf3aa16c-da81-4ead-a827-d136652992e6","resolution":{"observed_at":"2026-08-16T04:56:23.750792Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-22T06:32:14.747728+00:00","source":"crossref"},{"observed_at":"2026-08-22T06:32:06.552537+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":"2026-08-16T04:56:23.719484Z","title":"Szeg¨ o.Orthogonal polynomials, volume 23 of Amer","venue":null,"work_id":"1fa2f47e-db8b-4014-923d-8aa2b900f35c","year":1975},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":91,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:23.084188Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:3e6bc01dc86a25cad9492c71a05e19541c86ab31d0101805c37f435732cc20b2","observation_id":"57a7ca6c-16b8-4952-96ef-ce3948af36d7","resolution":{"observed_at":"2026-08-16T04:56:23.725996Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-22T06:32:14.747728+00:00","source":"crossref"},{"observed_at":"2026-08-22T06:32:06.552537+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":"2026-08-16T04:56:23.693154Z","title":"Greedy approximation","venue":null,"work_id":"f73c5ced-2e68-4fc8-ab9d-9e563a2b8eb5","year":2008},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":92,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:23.089911Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:335eba611a71ca8b7962bd92af3150406fdf0baa02240e681b71f1b42a588a8f","observation_id":"175afac9-2d64-43fd-b2a5-0f81ea3e9455","resolution":{"observed_at":"2026-08-16T04:56:23.701237Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-22T06:32:14.747728+00:00","source":"crossref"},{"observed_at":"2026-08-22T06:32:06.552537+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":"2026-08-16T04:56:23.662400Z","title":"Wainwright","venue":null,"work_id":"8cc0032b-31f8-4c91-89d6-d88f0b8f86fd","year":2019},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":93,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:23.095126Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:03e2eaa667219fc84c3612a67608189dc4be502818faed40aaee23ded357a42b","observation_id":"3022604c-407d-486e-8196-42b7ed935e0c","resolution":{"observed_at":"2026-08-16T04:56:23.669149Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-22T06:32:14.747728+00:00","source":"crossref"},{"observed_at":"2026-08-22T06:32:06.552537+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":"2026-08-16T04:56:23.635995Z","title":"An extreme learning machine-bas ed method for computa- tional pdes in higher dimensions","venue":null,"work_id":"c830acea-d776-4e53-b98d-41221daf83ce","year":2024},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":94,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:23.101735Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:457cb7d8de8788b3ac9128df9e6f39209a6ec631e261635b62d391c8ff4157d9","observation_id":"eb9eca53-1962-48f3-ba06-1614d449425a","resolution":{"observed_at":"2026-08-16T04:56:23.644165Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-22T06:32:14.747728+00:00","source":"crossref"},{"observed_at":"2026-08-22T06:32:06.552537+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":"2026-08-16T04:56:23.600911Z","title":"Iterative methods by space decomposition and sub space correction","venue":null,"work_id":"531f56fa-8396-4438-910a-4578a8354792","year":1992},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":95,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:23.107863Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:36f09e275cf4453bec3345c050442a33e37bc6723b5f5344544881bcaf36b8f4","observation_id":"c8d1d622-0625-461a-948e-afe80a2ad969","resolution":{"observed_at":"2026-08-16T04:56:23.608704Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-22T06:32:14.747728+00:00","source":"crossref"},{"observed_at":"2026-08-22T06:32:06.552537+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":"2026-08-16T04:56:23.570404Z","title":"Finite neuron method and convergence analysis","venue":null,"work_id":"778977a1-b733-4aeb-a879-7726aefeb119","year":2020},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":96,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:23.115027Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:bff02288dfe7bb89f45a6f91030a28fd1aa9363ebe6af7c36fa415cc25ef2e6b","observation_id":"f9e539d1-b5bf-49e9-bdf7-bf575261f121","resolution":{"observed_at":"2026-08-16T04:56:23.578817Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-22T06:32:14.747728+00:00","source":"crossref"},{"observed_at":"2026-08-22T06:32:06.552537+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":{"arxiv_id":"2407.17763","last_updated":"2025-04-17T21:50:28Z","snapshot_observed_at":"2026-08-19T09:48:12.999811Z","submitted_at":"2024-07-25T04:38:17Z","title":"Randomized Greedy Algorithms for Neural Network Optimization","version":3},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2407.17763","snapshot_observed_at":"2026-08-16T04:56:23.129591Z","title":"Eﬃcient and provably convergent r andomized greedy algorithms for neural network optimization","venue":null,"work_id":null,"year":2024},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":97,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:23.129591Z"},"links":{"cited_paper":"/paper/2407.17763","citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:37cb643b65755477c1cc8ee4f00ecb74ebbd362def11bb259329eb4ba5287513","observation_id":"8c7607f7-9f2d-4656-ab2f-f3165c01d867","resolution":{"observed_at":"2026-08-16T04:56:23.129591Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-16T04:56:23.540036Z","title":"Optimal rates of approximat ion by shallow relu k neural networks and applications to nonparametric regression","venue":null,"work_id":"8e94dee9-6488-462f-ad68-aa01c769d56a","year":2024},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":98,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:23.142205Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:5873d9adc87e3dc2b2ae34d3095a995736542aebca7b6559ec70b2c7c2299081","observation_id":"5ede2d74-dbbb-4f3c-8d0c-1fb0701105fb","resolution":{"observed_at":"2026-08-16T04:56:23.549725Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-22T06:32:14.747728+00:00","source":"crossref"},{"observed_at":"2026-08-22T06:32:06.552537+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":"2026-08-16T04:56:23.501534Z","title":"Sup -norm approximation bounds for networks through probabilistic methods","venue":null,"work_id":"bda900a3-a8db-4ac2-ab33-6575b547d3ed","year":1995},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":99,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:23.151003Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:e37bdce364a28c33e88d176c33143a52d189ad671eeb73670389ec4fa1382c00","observation_id":"0d912c59-c073-4b97-b0d1-28d20040f1ee","resolution":{"observed_at":"2026-08-16T04:56:23.521724Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-22T06:32:14.747728+00:00","source":"crossref"},{"observed_at":"2026-08-22T06:32:06.552537+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":"2026-08-16T04:56:23.469754Z","title":"Trans ferable neural networks for partial diﬀerential equations","venue":null,"work_id":"953afce1-daba-4e40-b703-54d50e162573","year":2024},"citing_paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks","version":2},"reference_index":100,"source":"pdf_text","source_observed_at":"2026-08-16T04:56:23.171892Z"},"links":{"citing_paper":"/paper/2505.00351"},"observation_digest":"sha256:03b2a836e772905638addc8979ca596cf71b9d5ad342fa12fc6a827d9e976542","observation_id":"48e487e6-761f-4bab-a66b-f79a4a53df9c","resolution":{"observed_at":"2026-08-16T04:56:23.479117Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-22T06:32:14.747728+00:00","source":"crossref"},{"observed_at":"2026-08-22T06:32:06.552537+00:00","source":"retraction_watch"}],"state":"measured"}}],"paper":{"arxiv_id":"2505.00351","last_updated":"2025-05-12T07:42:42Z","latest_version":2,"primary_category":"math.NA","snapshot_observed_at":"2026-08-16T04:42:19.583536Z","submitted_at":"2025-05-01T06:50:41Z","title":"Integral Representations of Sobolev Spaces via ReLU$^k$ Activation Function and Optimal Error Estimates for Linearized Networks"},"reference_resolution":{"displayed":99,"state_counts":{"malformed_identifier":0,"metadata_mismatch":0,"parse_uncertain":0,"unresolved":25,"verified_exact":0,"verified_fuzzy":74},"total_outbound_references":99},"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":"2026-08-22T06:32:14.747728+00:00","source":"crossref"},{"observed_at":"2026-08-22T06:32:06.552537+00:00","source":"retraction_watch"}],"thesis":"As of 22 August 2026, this Paper Citation Record lists 99 of 99 outbound references and 10 inbound Pith citation observations for arXiv:2505.00351."}