{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:QH3ZIX3KOHZ3JIWR7TRCSEHIG7","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":"45154bfe4d067f36a2f07b52d9b673b64ea8168601d51e59eb25e4dcefcb5e2e","cross_cats_sorted":["cs.IT","math.IT","math.PR","math.ST","stat.ML","stat.TH"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2024-12-06T15:10:04Z","title_canon_sha256":"e3f909040a4e08e46ea9c6bf3fa9f41fafec2189d36d45a0aa4fd6b9f9df5ea3"},"schema_version":"1.0","source":{"id":"2412.05109","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2412.05109","created_at":"2026-06-03T01:05:43Z"},{"alias_kind":"arxiv_version","alias_value":"2412.05109v2","created_at":"2026-06-03T01:05:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2412.05109","created_at":"2026-06-03T01:05:43Z"},{"alias_kind":"pith_short_12","alias_value":"QH3ZIX3KOHZ3","created_at":"2026-06-03T01:05:43Z"},{"alias_kind":"pith_short_16","alias_value":"QH3ZIX3KOHZ3JIWR","created_at":"2026-06-03T01:05:43Z"},{"alias_kind":"pith_short_8","alias_value":"QH3ZIX3K","created_at":"2026-06-03T01:05:43Z"}],"graph_snapshots":[{"event_id":"sha256:b2cd63f764102305997916e070ee7bb6460a43057b150a7038844297af282f01","target":"graph","created_at":"2026-06-03T01:05:43Z","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/2412.05109/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We derive universal approximation results for the class of (countably) $m$-rectifiable measures. Specifically, we prove that $m$-rectifiable measures can be approximated as push-forwards of the one-dimensional Lebesgue measure on $[0,1]$ using ReLU neural networks with arbitrarily small approximation error in terms of Wasserstein distance. What is more, the weights in the networks under consideration are quantized and bounded and the number of ReLU neural networks required to achieve an approximation error of $\\varepsilon$ is no larger than $2^{b(\\varepsilon)}$ with $b(\\varepsilon)=\\mathcal{O}","authors_text":"Alex B\\\"uhler, Erwin Riegler, Helmut B\\\"olcskei, Yang Pan","cross_cats":["cs.IT","math.IT","math.PR","math.ST","stat.ML","stat.TH"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2024-12-06T15:10:04Z","title":"Generating Rectifiable Measures through Neural Networks"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2412.05109","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:b9c48ca81b8e5ae7a7d4f49e24bf87eb5b9adb71eea632eba7ed2d2422a4c57e","target":"record","created_at":"2026-06-03T01:05:43Z","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":"45154bfe4d067f36a2f07b52d9b673b64ea8168601d51e59eb25e4dcefcb5e2e","cross_cats_sorted":["cs.IT","math.IT","math.PR","math.ST","stat.ML","stat.TH"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2024-12-06T15:10:04Z","title_canon_sha256":"e3f909040a4e08e46ea9c6bf3fa9f41fafec2189d36d45a0aa4fd6b9f9df5ea3"},"schema_version":"1.0","source":{"id":"2412.05109","kind":"arxiv","version":2}},"canonical_sha256":"81f7945f6a71f3b4a2d1fce22910e837c1851891a621922fde7fc1d648b6f132","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"81f7945f6a71f3b4a2d1fce22910e837c1851891a621922fde7fc1d648b6f132","first_computed_at":"2026-06-03T01:05:43.174726Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-03T01:05:43.174726Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"jqfoREgTQdt2nIki/YsxSuz6Oef4ylVyTCCKT+/khLQYE18fvo9KpMxo5aprTX6erB4HE2kydBRA+YGQX+dyCQ==","signature_status":"signed_v1","signed_at":"2026-06-03T01:05:43.175202Z","signed_message":"canonical_sha256_bytes"},"source_id":"2412.05109","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b9c48ca81b8e5ae7a7d4f49e24bf87eb5b9adb71eea632eba7ed2d2422a4c57e","sha256:b2cd63f764102305997916e070ee7bb6460a43057b150a7038844297af282f01"],"state_sha256":"988440a3ec0c5351ffc74ff8b1ddc40606362a05a138d88a06e26ea0cb36c51f"}