{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2017:APMC4X36ZKNF5QLIPLSIOIGAAG","short_pith_number":"pith:APMC4X36","schema_version":"1.0","canonical_sha256":"03d82e5f7eca9a5ec1687ae48720c001a19bfbffb1cdc2220ea02ee7205ef080","source":{"kind":"arxiv","id":"1705.01714","version":4},"attestation_state":"computed","paper":{"title":"Optimal Approximation with Sparsely Connected Deep Neural Networks","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.IT","math.FA","math.IT"],"primary_cat":"cs.LG","authors_text":"Gitta Kutyniok, Helmut B\\\"olcskei, Philipp Grohs, Philipp Petersen","submitted_at":"2017-05-04T06:45:06Z","abstract_excerpt":"We derive fundamental lower bounds on the connectivity and the memory requirements of deep neural networks guaranteeing uniform approximation rates for arbitrary function classes in $L^2(\\mathbb R^d)$. In other words, we establish a connection between the complexity of a function class and the complexity of deep neural networks approximating functions from this class to within a prescribed accuracy. Additionally, we prove that our lower bounds are achievable for a broad family of function classes. Specifically, all function classes that are optimally approximated by a general class of represen"},"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":"1705.01714","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2017-05-04T06:45:06Z","cross_cats_sorted":["cs.IT","math.FA","math.IT"],"title_canon_sha256":"e9d9fc7331c21a258dad7bf298f611525a7ead963bba3068d952b4a9198005ba","abstract_canon_sha256":"667bada4cbbed87854f902e0a593ea088442203a87c3025f54573bb2add644ae"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:15:52.589170Z","signature_b64":"fObc67xA2Fq8r1MBJ3y7qPUhoapnGjX0eFfZYRnxmrrM09zISSrqwFBm2kU3izEtSnUBK9AmfRbX0tzriPhrCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"03d82e5f7eca9a5ec1687ae48720c001a19bfbffb1cdc2220ea02ee7205ef080","last_reissued_at":"2026-05-18T00:15:52.588526Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:15:52.588526Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Optimal Approximation with Sparsely Connected Deep Neural Networks","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.IT","math.FA","math.IT"],"primary_cat":"cs.LG","authors_text":"Gitta Kutyniok, Helmut B\\\"olcskei, Philipp Grohs, Philipp Petersen","submitted_at":"2017-05-04T06:45:06Z","abstract_excerpt":"We derive fundamental lower bounds on the connectivity and the memory requirements of deep neural networks guaranteeing uniform approximation rates for arbitrary function classes in $L^2(\\mathbb R^d)$. In other words, we establish a connection between the complexity of a function class and the complexity of deep neural networks approximating functions from this class to within a prescribed accuracy. Additionally, we prove that our lower bounds are achievable for a broad family of function classes. Specifically, all function classes that are optimally approximated by a general class of represen"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1705.01714","kind":"arxiv","version":4},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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":"1705.01714","created_at":"2026-05-18T00:15:52.588616+00:00"},{"alias_kind":"arxiv_version","alias_value":"1705.01714v4","created_at":"2026-05-18T00:15:52.588616+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1705.01714","created_at":"2026-05-18T00:15:52.588616+00:00"},{"alias_kind":"pith_short_12","alias_value":"APMC4X36ZKNF","created_at":"2026-05-18T12:31:05.417338+00:00"},{"alias_kind":"pith_short_16","alias_value":"APMC4X36ZKNF5QLI","created_at":"2026-05-18T12:31:05.417338+00:00"},{"alias_kind":"pith_short_8","alias_value":"APMC4X36","created_at":"2026-05-18T12:31:05.417338+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/APMC4X36ZKNF5QLIPLSIOIGAAG","json":"https://pith.science/pith/APMC4X36ZKNF5QLIPLSIOIGAAG.json","graph_json":"https://pith.science/api/pith-number/APMC4X36ZKNF5QLIPLSIOIGAAG/graph.json","events_json":"https://pith.science/api/pith-number/APMC4X36ZKNF5QLIPLSIOIGAAG/events.json","paper":"https://pith.science/paper/APMC4X36"},"agent_actions":{"view_html":"https://pith.science/pith/APMC4X36ZKNF5QLIPLSIOIGAAG","download_json":"https://pith.science/pith/APMC4X36ZKNF5QLIPLSIOIGAAG.json","view_paper":"https://pith.science/paper/APMC4X36","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1705.01714&json=true","fetch_graph":"https://pith.science/api/pith-number/APMC4X36ZKNF5QLIPLSIOIGAAG/graph.json","fetch_events":"https://pith.science/api/pith-number/APMC4X36ZKNF5QLIPLSIOIGAAG/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/APMC4X36ZKNF5QLIPLSIOIGAAG/action/timestamp_anchor","attest_storage":"https://pith.science/pith/APMC4X36ZKNF5QLIPLSIOIGAAG/action/storage_attestation","attest_author":"https://pith.science/pith/APMC4X36ZKNF5QLIPLSIOIGAAG/action/author_attestation","sign_citation":"https://pith.science/pith/APMC4X36ZKNF5QLIPLSIOIGAAG/action/citation_signature","submit_replication":"https://pith.science/pith/APMC4X36ZKNF5QLIPLSIOIGAAG/action/replication_record"}},"created_at":"2026-05-18T00:15:52.588616+00:00","updated_at":"2026-05-18T00:15:52.588616+00:00"}