{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:J6R5L6GONWSF55M33GKJVYF27Y","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":"d61b09058000b6f724c67b9171f587d15d49a5d5ba44153170ce82c09ec0ed5d","cross_cats_sorted":["math.PR","math.ST","stat.ML","stat.TH"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2018-11-30T13:22:01Z","title_canon_sha256":"b2549d3a459bcab0b555e19bfd594088d3b4a4becfc52509853c6df5707c5ab8"},"schema_version":"1.0","source":{"id":"1811.12783","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1811.12783","created_at":"2026-05-17T23:59:28Z"},{"alias_kind":"arxiv_version","alias_value":"1811.12783v1","created_at":"2026-05-17T23:59:28Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1811.12783","created_at":"2026-05-17T23:59:28Z"},{"alias_kind":"pith_short_12","alias_value":"J6R5L6GONWSF","created_at":"2026-05-18T12:32:31Z"},{"alias_kind":"pith_short_16","alias_value":"J6R5L6GONWSF55M3","created_at":"2026-05-18T12:32:31Z"},{"alias_kind":"pith_short_8","alias_value":"J6R5L6GO","created_at":"2026-05-18T12:32:31Z"}],"graph_snapshots":[{"event_id":"sha256:8bc4a77c493a4a33c53d6002836d4dc88d33397358157695c51fb07c3c032a7e","target":"graph","created_at":"2026-05-17T23:59:28Z","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"},"paper":{"abstract_excerpt":"We present a formal measure-theoretical theory of neural networks (NN) built on probability coupling theory. Our main contributions are summarized as follows.\n  * Built on the formalism of probability coupling theory, we derive an algorithm framework, named Hierarchical Measure Group and Approximate System (HMGAS), nicknamed S-System, that is designed to learn the complex hierarchical, statistical dependency in the physical world.\n  * We show that NNs are special cases of S-System when the probability kernels assume certain exponential family distributions. Activation Functions are derived for","authors_text":"Shuai Li","cross_cats":["math.PR","math.ST","stat.ML","stat.TH"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2018-11-30T13:22:01Z","title":"Measure, Manifold, Learning, and Optimization: A Theory Of Neural Networks"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1811.12783","kind":"arxiv","version":1},"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":"sha256:5ea738cf7df292dea4bf6c348acb378c8a8cc08a1e43f60ca1835d633d39357b","target":"record","created_at":"2026-05-17T23:59:28Z","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":"d61b09058000b6f724c67b9171f587d15d49a5d5ba44153170ce82c09ec0ed5d","cross_cats_sorted":["math.PR","math.ST","stat.ML","stat.TH"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2018-11-30T13:22:01Z","title_canon_sha256":"b2549d3a459bcab0b555e19bfd594088d3b4a4becfc52509853c6df5707c5ab8"},"schema_version":"1.0","source":{"id":"1811.12783","kind":"arxiv","version":1}},"canonical_sha256":"4fa3d5f8ce6da45ef59bd9949ae0bafe3a4d89c8119f6bc3700d1acb97eca266","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4fa3d5f8ce6da45ef59bd9949ae0bafe3a4d89c8119f6bc3700d1acb97eca266","first_computed_at":"2026-05-17T23:59:28.475695Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:59:28.475695Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"tSKLJskPVdlGC0IJfFl1AJHDn3baR9YIbxX1heOtMFbjkIfqU1jUj7e22vweN9p0wOXFGVBAG3+6N2TnySPBDw==","signature_status":"signed_v1","signed_at":"2026-05-17T23:59:28.476203Z","signed_message":"canonical_sha256_bytes"},"source_id":"1811.12783","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5ea738cf7df292dea4bf6c348acb378c8a8cc08a1e43f60ca1835d633d39357b","sha256:8bc4a77c493a4a33c53d6002836d4dc88d33397358157695c51fb07c3c032a7e"],"state_sha256":"7310cd4b33ccc837f869c22c5ea808eeb9afd6cb50af7d0f73d9fa783cfaa1f6"}