{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2019:ZPI6IUKRB52M265BKU5XQY4G2D","short_pith_number":"pith:ZPI6IUKR","schema_version":"1.0","canonical_sha256":"cbd1e451510f74cd7ba1553b786386d0dc28ec40d9f30c0c0f27b2d0f7a24cc0","source":{"kind":"arxiv","id":"1908.00695","version":1},"attestation_state":"computed","paper":{"title":"Deep ReLU network approximation of functions on a manifold","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.LG"],"primary_cat":"stat.ML","authors_text":"Johannes Schmidt-Hieber","submitted_at":"2019-08-02T04:01:13Z","abstract_excerpt":"Whereas recovery of the manifold from data is a well-studied topic, approximation rates for functions defined on manifolds are less known. In this work, we study a regression problem with inputs on a $d^*$-dimensional manifold that is embedded into a space with potentially much larger ambient dimension. It is shown that sparsely connected deep ReLU networks can approximate a H\\\"older function with smoothness index $\\beta$ up to error $\\epsilon$ using of the order of $\\epsilon^{-d^*/\\beta}\\log(1/\\epsilon)$ many non-zero network parameters. As an application, we derive statistical convergence ra"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1908.00695","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"stat.ML","submitted_at":"2019-08-02T04:01:13Z","cross_cats_sorted":["cs.LG"],"title_canon_sha256":"cae8b0cb85fbafd1bb43b818f984237348d8d6b0baa1d970f7774120425b35c8","abstract_canon_sha256":"380efd4ab8aad83ed493875f316fb2b724c0d076e7e95451354d0e5b8a2c0cad"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T23:51:08.590421Z","signature_b64":"oqJNDQ2xRZgYCuHiGwYCsuI9btne0DYfX3jBXUCsm7KWbnQkVoTDbbzwIEWPCNXYCJmZL8JShD/uf2uBByG+Cg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cbd1e451510f74cd7ba1553b786386d0dc28ec40d9f30c0c0f27b2d0f7a24cc0","last_reissued_at":"2026-07-04T23:51:08.589880Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T23:51:08.589880Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Deep ReLU network approximation of functions on a manifold","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.LG"],"primary_cat":"stat.ML","authors_text":"Johannes Schmidt-Hieber","submitted_at":"2019-08-02T04:01:13Z","abstract_excerpt":"Whereas recovery of the manifold from data is a well-studied topic, approximation rates for functions defined on manifolds are less known. In this work, we study a regression problem with inputs on a $d^*$-dimensional manifold that is embedded into a space with potentially much larger ambient dimension. It is shown that sparsely connected deep ReLU networks can approximate a H\\\"older function with smoothness index $\\beta$ up to error $\\epsilon$ using of the order of $\\epsilon^{-d^*/\\beta}\\log(1/\\epsilon)$ many non-zero network parameters. As an application, we derive statistical convergence ra"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.00695","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1908.00695/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1908.00695","created_at":"2026-07-04T23:51:08.589934+00:00"},{"alias_kind":"arxiv_version","alias_value":"1908.00695v1","created_at":"2026-07-04T23:51:08.589934+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1908.00695","created_at":"2026-07-04T23:51:08.589934+00:00"},{"alias_kind":"pith_short_12","alias_value":"ZPI6IUKRB52M","created_at":"2026-07-04T23:51:08.589934+00:00"},{"alias_kind":"pith_short_16","alias_value":"ZPI6IUKRB52M265B","created_at":"2026-07-04T23:51:08.589934+00:00"},{"alias_kind":"pith_short_8","alias_value":"ZPI6IUKR","created_at":"2026-07-04T23:51:08.589934+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":4,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2606.08799","citing_title":"Generalization in Nonlinear Least Squares via Learned Feature Geometry","ref_index":76,"is_internal_anchor":false},{"citing_arxiv_id":"2505.03205","citing_title":"Transformers for Learning on Noisy and Task-Level Manifolds: Approximation and Generalization Insights","ref_index":11,"is_internal_anchor":false},{"citing_arxiv_id":"2506.10959","citing_title":"Understanding In-Context Learning on Structured Manifolds: Bridging Attention to Kernel Methods","ref_index":10,"is_internal_anchor":false},{"citing_arxiv_id":"2605.11558","citing_title":"A Composite Activation Function for Learning Stable Binary Representations","ref_index":60,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/ZPI6IUKRB52M265BKU5XQY4G2D","json":"https://pith.science/pith/ZPI6IUKRB52M265BKU5XQY4G2D.json","graph_json":"https://pith.science/api/pith-number/ZPI6IUKRB52M265BKU5XQY4G2D/graph.json","events_json":"https://pith.science/api/pith-number/ZPI6IUKRB52M265BKU5XQY4G2D/events.json","paper":"https://pith.science/paper/ZPI6IUKR"},"agent_actions":{"view_html":"https://pith.science/pith/ZPI6IUKRB52M265BKU5XQY4G2D","download_json":"https://pith.science/pith/ZPI6IUKRB52M265BKU5XQY4G2D.json","view_paper":"https://pith.science/paper/ZPI6IUKR","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1908.00695&json=true","fetch_graph":"https://pith.science/api/pith-number/ZPI6IUKRB52M265BKU5XQY4G2D/graph.json","fetch_events":"https://pith.science/api/pith-number/ZPI6IUKRB52M265BKU5XQY4G2D/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ZPI6IUKRB52M265BKU5XQY4G2D/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ZPI6IUKRB52M265BKU5XQY4G2D/action/storage_attestation","attest_author":"https://pith.science/pith/ZPI6IUKRB52M265BKU5XQY4G2D/action/author_attestation","sign_citation":"https://pith.science/pith/ZPI6IUKRB52M265BKU5XQY4G2D/action/citation_signature","submit_replication":"https://pith.science/pith/ZPI6IUKRB52M265BKU5XQY4G2D/action/replication_record"}},"created_at":"2026-07-04T23:51:08.589934+00:00","updated_at":"2026-07-04T23:51:08.589934+00:00"}