{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2023:WJXU74SUC7CVVVFCW67MNW2ASG","short_pith_number":"pith:WJXU74SU","canonical_record":{"source":{"id":"2304.12231","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2023-04-24T16:18:22Z","cross_cats_sorted":["cs.NA","cs.NE","math.NA","math.PR","stat.ML"],"title_canon_sha256":"8623abad6a39500f0a6a3b66ce6b272e16934814238c2decd3d1ad6489440c45","abstract_canon_sha256":"4e1476d50c7228864f0516b146716a2209edbab4632a55083df5ec91e013a695"},"schema_version":"1.0"},"canonical_sha256":"b26f4ff25417c55ad4a2b7bec6db4091bae89e40d17e95a9412d8b837dd60d21","source":{"kind":"arxiv","id":"2304.12231","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2304.12231","created_at":"2026-07-05T06:33:45Z"},{"alias_kind":"arxiv_version","alias_value":"2304.12231v2","created_at":"2026-07-05T06:33:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2304.12231","created_at":"2026-07-05T06:33:45Z"},{"alias_kind":"pith_short_12","alias_value":"WJXU74SUC7CV","created_at":"2026-07-05T06:33:45Z"},{"alias_kind":"pith_short_16","alias_value":"WJXU74SUC7CVVVFC","created_at":"2026-07-05T06:33:45Z"},{"alias_kind":"pith_short_8","alias_value":"WJXU74SU","created_at":"2026-07-05T06:33:45Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2023:WJXU74SUC7CVVVFCW67MNW2ASG","target":"record","payload":{"canonical_record":{"source":{"id":"2304.12231","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2023-04-24T16:18:22Z","cross_cats_sorted":["cs.NA","cs.NE","math.NA","math.PR","stat.ML"],"title_canon_sha256":"8623abad6a39500f0a6a3b66ce6b272e16934814238c2decd3d1ad6489440c45","abstract_canon_sha256":"4e1476d50c7228864f0516b146716a2209edbab4632a55083df5ec91e013a695"},"schema_version":"1.0"},"canonical_sha256":"b26f4ff25417c55ad4a2b7bec6db4091bae89e40d17e95a9412d8b837dd60d21","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T06:33:45.199958Z","signature_b64":"qCTR4Rlp96WzTbf7a22BeZ4Z5auulFH9OHUWshrOCyiraXDehmbGZDJkx7TiOUg43nsFtpoXVWnFoBbivLyACQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b26f4ff25417c55ad4a2b7bec6db4091bae89e40d17e95a9412d8b837dd60d21","last_reissued_at":"2026-07-05T06:33:45.199490Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T06:33:45.199490Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2304.12231","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T06:33:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"tvg0A/llYWm9ThHHCCXsM1trsYxkh8cY6bnFwJK1BC8ZPOHbHfsANRCAKyo7IvDITsww5JlsAtA0V8pJ7qSZBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-09T20:52:32.745798Z"},"content_sha256":"2a52f060806cc4c3ad6c1376c60803d00815aafcb66b92bf42eb2879b4e9c46e","schema_version":"1.0","event_id":"sha256:2a52f060806cc4c3ad6c1376c60803d00815aafcb66b92bf42eb2879b4e9c46e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2023:WJXU74SUC7CVVVFCW67MNW2ASG","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"An Approximation Theory for Metric Space-Valued Functions With A View Towards Deep Learning","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.NA","cs.NE","math.NA","math.PR","stat.ML"],"primary_cat":"cs.LG","authors_text":"Anastasis Kratsios, Chong Liu, Ivan Dokmani\\'c, Maarten V. de Hoop, Matti Lassas","submitted_at":"2023-04-24T16:18:22Z","abstract_excerpt":"Motivated by the developing mathematics of deep learning, we build universal functions approximators of continuous maps between arbitrary Polish metric spaces $\\mathcal{X}$ and $\\mathcal{Y}$ using elementary functions between Euclidean spaces as building blocks. Earlier results assume that the target space $\\mathcal{Y}$ is a topological vector space. We overcome this limitation by ``randomization'': our approximators output discrete probability measures over $\\mathcal{Y}$. When $\\mathcal{X}$ and $\\mathcal{Y}$ are Polish without additional structure, we prove very general qualitative guarantees"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2304.12231","kind":"arxiv","version":2},"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/2304.12231/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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T06:33:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"H6oAH11IBx/pP05JJhcU9QT5vvrxxSe27PLBO7U94IB/Hl5Y25zvwXg4HmDR7LZNqMmoHazS030nsOlhQJsIDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-09T20:52:32.746300Z"},"content_sha256":"f1d0c8cfa6cb6f15d3ffe1212018390f15cede8b66d6dbfd1f670b8feb360245","schema_version":"1.0","event_id":"sha256:f1d0c8cfa6cb6f15d3ffe1212018390f15cede8b66d6dbfd1f670b8feb360245"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/WJXU74SUC7CVVVFCW67MNW2ASG/bundle.json","state_url":"https://pith.science/pith/WJXU74SUC7CVVVFCW67MNW2ASG/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/WJXU74SUC7CVVVFCW67MNW2ASG/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-08-09T20:52:32Z","links":{"resolver":"https://pith.science/pith/WJXU74SUC7CVVVFCW67MNW2ASG","bundle":"https://pith.science/pith/WJXU74SUC7CVVVFCW67MNW2ASG/bundle.json","state":"https://pith.science/pith/WJXU74SUC7CVVVFCW67MNW2ASG/state.json","well_known_bundle":"https://pith.science/.well-known/pith/WJXU74SUC7CVVVFCW67MNW2ASG/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2023:WJXU74SUC7CVVVFCW67MNW2ASG","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":"4e1476d50c7228864f0516b146716a2209edbab4632a55083df5ec91e013a695","cross_cats_sorted":["cs.NA","cs.NE","math.NA","math.PR","stat.ML"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2023-04-24T16:18:22Z","title_canon_sha256":"8623abad6a39500f0a6a3b66ce6b272e16934814238c2decd3d1ad6489440c45"},"schema_version":"1.0","source":{"id":"2304.12231","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2304.12231","created_at":"2026-07-05T06:33:45Z"},{"alias_kind":"arxiv_version","alias_value":"2304.12231v2","created_at":"2026-07-05T06:33:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2304.12231","created_at":"2026-07-05T06:33:45Z"},{"alias_kind":"pith_short_12","alias_value":"WJXU74SUC7CV","created_at":"2026-07-05T06:33:45Z"},{"alias_kind":"pith_short_16","alias_value":"WJXU74SUC7CVVVFC","created_at":"2026-07-05T06:33:45Z"},{"alias_kind":"pith_short_8","alias_value":"WJXU74SU","created_at":"2026-07-05T06:33:45Z"}],"graph_snapshots":[{"event_id":"sha256:f1d0c8cfa6cb6f15d3ffe1212018390f15cede8b66d6dbfd1f670b8feb360245","target":"graph","created_at":"2026-07-05T06:33:45Z","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/2304.12231/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Motivated by the developing mathematics of deep learning, we build universal functions approximators of continuous maps between arbitrary Polish metric spaces $\\mathcal{X}$ and $\\mathcal{Y}$ using elementary functions between Euclidean spaces as building blocks. Earlier results assume that the target space $\\mathcal{Y}$ is a topological vector space. We overcome this limitation by ``randomization'': our approximators output discrete probability measures over $\\mathcal{Y}$. When $\\mathcal{X}$ and $\\mathcal{Y}$ are Polish without additional structure, we prove very general qualitative guarantees","authors_text":"Anastasis Kratsios, Chong Liu, Ivan Dokmani\\'c, Maarten V. de Hoop, Matti Lassas","cross_cats":["cs.NA","cs.NE","math.NA","math.PR","stat.ML"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2023-04-24T16:18:22Z","title":"An Approximation Theory for Metric Space-Valued Functions With A View Towards Deep Learning"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2304.12231","kind":"arxiv","version":2},"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":"sha256:2a52f060806cc4c3ad6c1376c60803d00815aafcb66b92bf42eb2879b4e9c46e","target":"record","created_at":"2026-07-05T06:33:45Z","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":"4e1476d50c7228864f0516b146716a2209edbab4632a55083df5ec91e013a695","cross_cats_sorted":["cs.NA","cs.NE","math.NA","math.PR","stat.ML"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2023-04-24T16:18:22Z","title_canon_sha256":"8623abad6a39500f0a6a3b66ce6b272e16934814238c2decd3d1ad6489440c45"},"schema_version":"1.0","source":{"id":"2304.12231","kind":"arxiv","version":2}},"canonical_sha256":"b26f4ff25417c55ad4a2b7bec6db4091bae89e40d17e95a9412d8b837dd60d21","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b26f4ff25417c55ad4a2b7bec6db4091bae89e40d17e95a9412d8b837dd60d21","first_computed_at":"2026-07-05T06:33:45.199490Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T06:33:45.199490Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"qCTR4Rlp96WzTbf7a22BeZ4Z5auulFH9OHUWshrOCyiraXDehmbGZDJkx7TiOUg43nsFtpoXVWnFoBbivLyACQ==","signature_status":"signed_v1","signed_at":"2026-07-05T06:33:45.199958Z","signed_message":"canonical_sha256_bytes"},"source_id":"2304.12231","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2a52f060806cc4c3ad6c1376c60803d00815aafcb66b92bf42eb2879b4e9c46e","sha256:f1d0c8cfa6cb6f15d3ffe1212018390f15cede8b66d6dbfd1f670b8feb360245"],"state_sha256":"046050f6000fd6f8c5020cf45f76849955b5e367c20c341f3d4f23c53c7d1d44"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3YRygP/FD05500luZY8pqtnfhxqrjTa8YGy3uYq+WztrbZwVNXFdtjcjXaMDLoMXezOZpqp0gdpgSV+1Fe2zDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-09T20:52:32.749748Z","bundle_sha256":"61d59e1f5a518aae82d35f1503958f6ec496846260b8009c824d4a55087fc085"}}