{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2023:LTYWVBHGSKNPSRPTN7TPBWGXQ5","short_pith_number":"pith:LTYWVBHG","canonical_record":{"source":{"id":"2309.04644","kind":"arxiv","version":3},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2023-09-09T00:05:45Z","cross_cats_sorted":[],"title_canon_sha256":"6e2cbb0d779798373288c835a31ed6b41b632b225d4d5129a480cbb70c15f983","abstract_canon_sha256":"9e6d17d143fbc8daca03d3a5c96f38d43f762f18756a4a1d43a39d582ba5c383"},"schema_version":"1.0"},"canonical_sha256":"5cf16a84e6929af945f36fe6f0d8d7877bc0c714840ae357b60405f7289e084b","source":{"kind":"arxiv","id":"2309.04644","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2309.04644","created_at":"2026-07-05T09:03:16Z"},{"alias_kind":"arxiv_version","alias_value":"2309.04644v3","created_at":"2026-07-05T09:03:16Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2309.04644","created_at":"2026-07-05T09:03:16Z"},{"alias_kind":"pith_short_12","alias_value":"LTYWVBHGSKNP","created_at":"2026-07-05T09:03:16Z"},{"alias_kind":"pith_short_16","alias_value":"LTYWVBHGSKNPSRPT","created_at":"2026-07-05T09:03:16Z"},{"alias_kind":"pith_short_8","alias_value":"LTYWVBHG","created_at":"2026-07-05T09:03:16Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2023:LTYWVBHGSKNPSRPTN7TPBWGXQ5","target":"record","payload":{"canonical_record":{"source":{"id":"2309.04644","kind":"arxiv","version":3},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2023-09-09T00:05:45Z","cross_cats_sorted":[],"title_canon_sha256":"6e2cbb0d779798373288c835a31ed6b41b632b225d4d5129a480cbb70c15f983","abstract_canon_sha256":"9e6d17d143fbc8daca03d3a5c96f38d43f762f18756a4a1d43a39d582ba5c383"},"schema_version":"1.0"},"canonical_sha256":"5cf16a84e6929af945f36fe6f0d8d7877bc0c714840ae357b60405f7289e084b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:03:16.430938Z","signature_b64":"Q9opdIzcJkX6yLkBlgxvondTMAe1iQTTLVto35BfsyL0pWwn/Jid/IXoeKApRf2sffCMqN/qUh2nBjVeGtKJBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5cf16a84e6929af945f36fe6f0d8d7877bc0c714840ae357b60405f7289e084b","last_reissued_at":"2026-07-05T09:03:16.430538Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:03:16.430538Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2309.04644","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T09:03:16Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"yW9WAVM4MGP0t8VpBxsAVt7nV5q6HTdeStwruoRzIvm15UP8jMlKGBGFiVmkI6dmZcdGMFHFcPal+nVWhyUUCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-18T11:53:35.654509Z"},"content_sha256":"0149d159c448a5b70a8a6dee0202147cc16687bf907199024bdb7b4e4bbb5d3a","schema_version":"1.0","event_id":"sha256:0149d159c448a5b70a8a6dee0202147cc16687bf907199024bdb7b4e4bbb5d3a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2023:LTYWVBHGSKNPSRPTN7TPBWGXQ5","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Towards Understanding Neural Collapse: The Effects of Batch Normalization and Weight Decay","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"cs.LG","authors_text":"Leyan Pan, Xinyuan Cao","submitted_at":"2023-09-09T00:05:45Z","abstract_excerpt":"Neural Collapse (NC) is a geometric structure recently observed at the terminal phase of training deep neural networks, which states that last-layer feature vectors for the same class would \"collapse\" to a single point, while features of different classes become equally separated. We demonstrate that batch normalization (BN) and weight decay (WD) critically influence the emergence of NC. In the near-optimal loss regime, we establish an asymptotic lower bound on the emergence of NC that depends only on the WD value, training loss, and the presence of last-layer BN. Our experiments substantiate "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2309.04644","kind":"arxiv","version":3},"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/2309.04644/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":"2026-07-05T09:03:16Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"g6pl0r6Q5V8az/OlS6hw7uCYR84eVrrlmozq9dbuk1Z3OOzCO/hkniDC06X2OuYD+GlCjWedfYJEKaZ+t3H7CQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-18T11:53:35.654882Z"},"content_sha256":"18046bd44579c1f99bc363f4fe889f5b3c575de62c88666581525179856ad758","schema_version":"1.0","event_id":"sha256:18046bd44579c1f99bc363f4fe889f5b3c575de62c88666581525179856ad758"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/LTYWVBHGSKNPSRPTN7TPBWGXQ5/bundle.json","state_url":"https://pith.science/pith/LTYWVBHGSKNPSRPTN7TPBWGXQ5/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/LTYWVBHGSKNPSRPTN7TPBWGXQ5/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":"2026-07-18T11:53:35Z","links":{"resolver":"https://pith.science/pith/LTYWVBHGSKNPSRPTN7TPBWGXQ5","bundle":"https://pith.science/pith/LTYWVBHGSKNPSRPTN7TPBWGXQ5/bundle.json","state":"https://pith.science/pith/LTYWVBHGSKNPSRPTN7TPBWGXQ5/state.json","well_known_bundle":"https://pith.science/.well-known/pith/LTYWVBHGSKNPSRPTN7TPBWGXQ5/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2023:LTYWVBHGSKNPSRPTN7TPBWGXQ5","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":"9e6d17d143fbc8daca03d3a5c96f38d43f762f18756a4a1d43a39d582ba5c383","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2023-09-09T00:05:45Z","title_canon_sha256":"6e2cbb0d779798373288c835a31ed6b41b632b225d4d5129a480cbb70c15f983"},"schema_version":"1.0","source":{"id":"2309.04644","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2309.04644","created_at":"2026-07-05T09:03:16Z"},{"alias_kind":"arxiv_version","alias_value":"2309.04644v3","created_at":"2026-07-05T09:03:16Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2309.04644","created_at":"2026-07-05T09:03:16Z"},{"alias_kind":"pith_short_12","alias_value":"LTYWVBHGSKNP","created_at":"2026-07-05T09:03:16Z"},{"alias_kind":"pith_short_16","alias_value":"LTYWVBHGSKNPSRPT","created_at":"2026-07-05T09:03:16Z"},{"alias_kind":"pith_short_8","alias_value":"LTYWVBHG","created_at":"2026-07-05T09:03:16Z"}],"graph_snapshots":[{"event_id":"sha256:18046bd44579c1f99bc363f4fe889f5b3c575de62c88666581525179856ad758","target":"graph","created_at":"2026-07-05T09:03:16Z","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/2309.04644/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Neural Collapse (NC) is a geometric structure recently observed at the terminal phase of training deep neural networks, which states that last-layer feature vectors for the same class would \"collapse\" to a single point, while features of different classes become equally separated. We demonstrate that batch normalization (BN) and weight decay (WD) critically influence the emergence of NC. In the near-optimal loss regime, we establish an asymptotic lower bound on the emergence of NC that depends only on the WD value, training loss, and the presence of last-layer BN. Our experiments substantiate ","authors_text":"Leyan Pan, Xinyuan Cao","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2023-09-09T00:05:45Z","title":"Towards Understanding Neural Collapse: The Effects of Batch Normalization and Weight Decay"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2309.04644","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":"sha256:0149d159c448a5b70a8a6dee0202147cc16687bf907199024bdb7b4e4bbb5d3a","target":"record","created_at":"2026-07-05T09:03:16Z","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":"9e6d17d143fbc8daca03d3a5c96f38d43f762f18756a4a1d43a39d582ba5c383","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2023-09-09T00:05:45Z","title_canon_sha256":"6e2cbb0d779798373288c835a31ed6b41b632b225d4d5129a480cbb70c15f983"},"schema_version":"1.0","source":{"id":"2309.04644","kind":"arxiv","version":3}},"canonical_sha256":"5cf16a84e6929af945f36fe6f0d8d7877bc0c714840ae357b60405f7289e084b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5cf16a84e6929af945f36fe6f0d8d7877bc0c714840ae357b60405f7289e084b","first_computed_at":"2026-07-05T09:03:16.430538Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T09:03:16.430538Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Q9opdIzcJkX6yLkBlgxvondTMAe1iQTTLVto35BfsyL0pWwn/Jid/IXoeKApRf2sffCMqN/qUh2nBjVeGtKJBw==","signature_status":"signed_v1","signed_at":"2026-07-05T09:03:16.430938Z","signed_message":"canonical_sha256_bytes"},"source_id":"2309.04644","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0149d159c448a5b70a8a6dee0202147cc16687bf907199024bdb7b4e4bbb5d3a","sha256:18046bd44579c1f99bc363f4fe889f5b3c575de62c88666581525179856ad758"],"state_sha256":"6ddcded539441e45b809362c16ef21e48b6f52d20fa42fa795012b0ad39a091a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9NQaaeX6mSPIrcBSoMBtYj93+IbF8tZ+hQanyzWDdThrqP4TkC//6SLfq/i/SpT7lp4ZYl7u3Nw15XSCokmTDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-18T11:53:35.657555Z","bundle_sha256":"602521d3b0ebd2db3be84d38d88106127113caa74fab4e3bcc9b0eeeb7934901"}}