{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:DM6646NCCOX2VBF47PYLNULRAG","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"a123194973cf940c993997c214243ecf0848182469b36034c35744b7dd1d64e8","cross_cats_sorted":["math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GN","submitted_at":"2018-06-22T00:30:51Z","title_canon_sha256":"25910798bc420ec248188a5eb617aa7f88783730ecf06a32fa5ad9c2ab2169ee"},"schema_version":"1.0","source":{"id":"1806.08459","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1806.08459","created_at":"2026-07-05T00:35:18Z"},{"alias_kind":"arxiv_version","alias_value":"1806.08459v3","created_at":"2026-07-05T00:35:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1806.08459","created_at":"2026-07-05T00:35:18Z"},{"alias_kind":"pith_short_12","alias_value":"DM6646NCCOX2","created_at":"2026-07-05T00:35:18Z"},{"alias_kind":"pith_short_16","alias_value":"DM6646NCCOX2VBF4","created_at":"2026-07-05T00:35:18Z"},{"alias_kind":"pith_short_8","alias_value":"DM6646NC","created_at":"2026-07-05T00:35:18Z"}],"graph_snapshots":[{"event_id":"sha256:50c62313e878a82c73c8b2e781a7568e927ee3b8922bf3a0118d48467c238021","target":"graph","created_at":"2026-07-05T00:35:18Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/1806.08459/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We analyze the topological properties of the set of functions that can be implemented by neural networks of a fixed size. Surprisingly, this set has many undesirable properties. It is highly non-convex, except possibly for a few exotic activation functions. Moreover, the set is not closed with respect to $L^p$-norms, $0 < p < \\infty$, for all practically-used activation functions, and also not closed with respect to the $L^\\infty$-norm for all practically-used activation functions except for the ReLU and the parametric ReLU. Finally, the function that maps a family of weights to the function c","authors_text":"Felix Voigtlaender, Mones Raslan, Philipp Petersen","cross_cats":["math.FA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GN","submitted_at":"2018-06-22T00:30:51Z","title":"Topological properties of the set of functions generated by neural networks of fixed size"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.08459","kind":"arxiv","version":3},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:722800de7154398441249c8991a6185505ab066cde35a04dc54fa157d0b5eb6d","target":"record","created_at":"2026-07-05T00:35:18Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"a123194973cf940c993997c214243ecf0848182469b36034c35744b7dd1d64e8","cross_cats_sorted":["math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GN","submitted_at":"2018-06-22T00:30:51Z","title_canon_sha256":"25910798bc420ec248188a5eb617aa7f88783730ecf06a32fa5ad9c2ab2169ee"},"schema_version":"1.0","source":{"id":"1806.08459","kind":"arxiv","version":3}},"canonical_sha256":"1b3dee79a213afaa84bcfbf0b6d171019ba36346df26528a947df6446060754c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1b3dee79a213afaa84bcfbf0b6d171019ba36346df26528a947df6446060754c","first_computed_at":"2026-07-05T00:35:18.094365Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T00:35:18.094365Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"HVOOtJfxXfkFQbsT8gd6tVpQB7vANlZurKP8aee99yKa3xP4Re0WIWNJCD1rAnyZSKm92kbEVnFES/++Jw9uDA==","signature_status":"signed_v1","signed_at":"2026-07-05T00:35:18.094834Z","signed_message":"canonical_sha256_bytes"},"source_id":"1806.08459","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:722800de7154398441249c8991a6185505ab066cde35a04dc54fa157d0b5eb6d","sha256:50c62313e878a82c73c8b2e781a7568e927ee3b8922bf3a0118d48467c238021"],"state_sha256":"cb99b111f3538beb3f7ed2f6f7f2d0d7553a3565085a5d23d21c6ae11cdd1540"}