{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:YPVXIHOC4CMWPKGAPJRJASAYKL","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":"cf6f472759ac7905d4430e0775bdf299f4be3ce32c824d34ee846cd25c28caf0","cross_cats_sorted":["nlin.AO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"q-bio.NC","submitted_at":"2011-02-01T14:32:38Z","title_canon_sha256":"8c25cb5e426b7778623ef38110fa99429b2acdc69d8f70b1fe1c250df98e64e8"},"schema_version":"1.0","source":{"id":"1102.0166","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1102.0166","created_at":"2026-05-18T04:30:13Z"},{"alias_kind":"arxiv_version","alias_value":"1102.0166v1","created_at":"2026-05-18T04:30:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1102.0166","created_at":"2026-05-18T04:30:13Z"},{"alias_kind":"pith_short_12","alias_value":"YPVXIHOC4CMW","created_at":"2026-05-18T12:26:47Z"},{"alias_kind":"pith_short_16","alias_value":"YPVXIHOC4CMWPKGA","created_at":"2026-05-18T12:26:47Z"},{"alias_kind":"pith_short_8","alias_value":"YPVXIHOC","created_at":"2026-05-18T12:26:47Z"}],"graph_snapshots":[{"event_id":"sha256:11dace7bc14a65f4d52a42cd87fc4e68d81ebbc5a8b65a72cd817cc5e3ec7acc","target":"graph","created_at":"2026-05-18T04:30:13Z","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 show how a Hopfield network with modifiable recurrent connections undergoing slow Hebbian learning can extract the underlying geometry of an input space. First, we use a slow/fast analysis to derive an averaged system whose dynamics derives from an energy function and therefore always converges to equilibrium points. The equilibria reflect the correlation structure of the inputs, a global object extracted through local recurrent interactions only. Second, we use numerical methods to illustrate how learning extracts the hidden geometrical structure of the inputs. Indeed, multidimensional sca","authors_text":"Mathieu N. Galtier, Olivier D. Faugeras, Paul C. Bressloff","cross_cats":["nlin.AO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"q-bio.NC","submitted_at":"2011-02-01T14:32:38Z","title":"Hebbian learning of recurrent connections: a geometrical perspective"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1102.0166","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:a5e24411aa0f20f749a1e9a1501a0f18ab8f76d9853b1fca7d09b3872b82bf5d","target":"record","created_at":"2026-05-18T04:30:13Z","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":"cf6f472759ac7905d4430e0775bdf299f4be3ce32c824d34ee846cd25c28caf0","cross_cats_sorted":["nlin.AO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"q-bio.NC","submitted_at":"2011-02-01T14:32:38Z","title_canon_sha256":"8c25cb5e426b7778623ef38110fa99429b2acdc69d8f70b1fe1c250df98e64e8"},"schema_version":"1.0","source":{"id":"1102.0166","kind":"arxiv","version":1}},"canonical_sha256":"c3eb741dc2e09967a8c07a6290481852e04899470da63ce17fe02f7224368cae","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c3eb741dc2e09967a8c07a6290481852e04899470da63ce17fe02f7224368cae","first_computed_at":"2026-05-18T04:30:13.049232Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:30:13.049232Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"bxgM5Ch6SC0kA1lPM5s0Jg3PiGNnoEMJDzbRKQ76Lq/tQ0VK8xBvo7z8KjD3dQnHMcBwHabBBxyzuwF+0U7DCg==","signature_status":"signed_v1","signed_at":"2026-05-18T04:30:13.049797Z","signed_message":"canonical_sha256_bytes"},"source_id":"1102.0166","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a5e24411aa0f20f749a1e9a1501a0f18ab8f76d9853b1fca7d09b3872b82bf5d","sha256:11dace7bc14a65f4d52a42cd87fc4e68d81ebbc5a8b65a72cd817cc5e3ec7acc"],"state_sha256":"a1134ae06ddd1923951946898239962089324afa527325fac594ab0cb5fc8570"}