{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2020:24KH5HCAVV3FO3KPCKU23LXBEN","short_pith_number":"pith:24KH5HCA","schema_version":"1.0","canonical_sha256":"d7147e9c40ad76576d4f12a9adaee1237c033b9910ea94016eeeb6c3eddf7a6c","source":{"kind":"arxiv","id":"2012.05420","version":3},"attestation_state":"computed","paper":{"title":"On the emergence of simplex symmetry in the final and penultimate layers of neural network classifiers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["stat.ML"],"primary_cat":"cs.LG","authors_text":"Stephan Wojtowytsch, Weinan E","submitted_at":"2020-12-10T02:32:52Z","abstract_excerpt":"A recent numerical study observed that neural network classifiers enjoy a large degree of symmetry in the penultimate layer. Namely, if $h(x) = Af(x) +b$ where $A$ is a linear map and $f$ is the output of the penultimate layer of the network (after activation), then all data points $x_{i, 1}, \\dots, x_{i, N_i}$ in a class $C_i$ are mapped to a single point $y_i$ by $f$ and the points $y_i$ are located at the vertices of a regular $k-1$-dimensional standard simplex in a high-dimensional Euclidean space.\n  We explain this observation analytically in toy models for highly expressive deep neural n"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2012.05420","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2020-12-10T02:32:52Z","cross_cats_sorted":["stat.ML"],"title_canon_sha256":"feff0465e941f4354be9ae83f9cb1976e21fe96b08271a29db9f4ea107312159","abstract_canon_sha256":"600fc86cb7f14f8d286476c900059f29278069fed54f04989b93d72cbe87b29e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T02:46:11.243010Z","signature_b64":"ES8uyVySWXn+B2KpdgG2cQq7XeuL75f3fldy7TCmSOh36EQ/FYPpNe7J/cA2DAVbOsw23ShZf5zgng0l9nkgDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d7147e9c40ad76576d4f12a9adaee1237c033b9910ea94016eeeb6c3eddf7a6c","last_reissued_at":"2026-07-05T02:46:11.242631Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T02:46:11.242631Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the emergence of simplex symmetry in the final and penultimate layers of neural network classifiers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["stat.ML"],"primary_cat":"cs.LG","authors_text":"Stephan Wojtowytsch, Weinan E","submitted_at":"2020-12-10T02:32:52Z","abstract_excerpt":"A recent numerical study observed that neural network classifiers enjoy a large degree of symmetry in the penultimate layer. Namely, if $h(x) = Af(x) +b$ where $A$ is a linear map and $f$ is the output of the penultimate layer of the network (after activation), then all data points $x_{i, 1}, \\dots, x_{i, N_i}$ in a class $C_i$ are mapped to a single point $y_i$ by $f$ and the points $y_i$ are located at the vertices of a regular $k-1$-dimensional standard simplex in a high-dimensional Euclidean space.\n  We explain this observation analytically in toy models for highly expressive deep neural n"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2012.05420","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/2012.05420/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2012.05420","created_at":"2026-07-05T02:46:11.242697+00:00"},{"alias_kind":"arxiv_version","alias_value":"2012.05420v3","created_at":"2026-07-05T02:46:11.242697+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2012.05420","created_at":"2026-07-05T02:46:11.242697+00:00"},{"alias_kind":"pith_short_12","alias_value":"24KH5HCAVV3F","created_at":"2026-07-05T02:46:11.242697+00:00"},{"alias_kind":"pith_short_16","alias_value":"24KH5HCAVV3FO3KP","created_at":"2026-07-05T02:46:11.242697+00:00"},{"alias_kind":"pith_short_8","alias_value":"24KH5HCA","created_at":"2026-07-05T02:46:11.242697+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2504.18437","citing_title":"Enhancing Pre-Trained Model-Based Class-Incremental Learning through Neural Collapse","ref_index":57,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/24KH5HCAVV3FO3KPCKU23LXBEN","json":"https://pith.science/pith/24KH5HCAVV3FO3KPCKU23LXBEN.json","graph_json":"https://pith.science/api/pith-number/24KH5HCAVV3FO3KPCKU23LXBEN/graph.json","events_json":"https://pith.science/api/pith-number/24KH5HCAVV3FO3KPCKU23LXBEN/events.json","paper":"https://pith.science/paper/24KH5HCA"},"agent_actions":{"view_html":"https://pith.science/pith/24KH5HCAVV3FO3KPCKU23LXBEN","download_json":"https://pith.science/pith/24KH5HCAVV3FO3KPCKU23LXBEN.json","view_paper":"https://pith.science/paper/24KH5HCA","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2012.05420&json=true","fetch_graph":"https://pith.science/api/pith-number/24KH5HCAVV3FO3KPCKU23LXBEN/graph.json","fetch_events":"https://pith.science/api/pith-number/24KH5HCAVV3FO3KPCKU23LXBEN/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/24KH5HCAVV3FO3KPCKU23LXBEN/action/timestamp_anchor","attest_storage":"https://pith.science/pith/24KH5HCAVV3FO3KPCKU23LXBEN/action/storage_attestation","attest_author":"https://pith.science/pith/24KH5HCAVV3FO3KPCKU23LXBEN/action/author_attestation","sign_citation":"https://pith.science/pith/24KH5HCAVV3FO3KPCKU23LXBEN/action/citation_signature","submit_replication":"https://pith.science/pith/24KH5HCAVV3FO3KPCKU23LXBEN/action/replication_record"}},"created_at":"2026-07-05T02:46:11.242697+00:00","updated_at":"2026-07-05T02:46:11.242697+00:00"}