{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:ZGM5JPI2CSG4PHJQ3MEJO6MWUL","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":"0bde633c669432bf4a29b4d650dfe008f34f855fa4594c66e29586c54a1a6c8a","cross_cats_sorted":["math.AG","stat.ML"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2019-10-03T18:22:30Z","title_canon_sha256":"f803ab99b86b7630266c9e406503ab59af0e6840809bebe63c2b76fad4fc7376"},"schema_version":"1.0","source":{"id":"1910.01671","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1910.01671","created_at":"2026-07-05T00:52:27Z"},{"alias_kind":"arxiv_version","alias_value":"1910.01671v2","created_at":"2026-07-05T00:52:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1910.01671","created_at":"2026-07-05T00:52:27Z"},{"alias_kind":"pith_short_12","alias_value":"ZGM5JPI2CSG4","created_at":"2026-07-05T00:52:27Z"},{"alias_kind":"pith_short_16","alias_value":"ZGM5JPI2CSG4PHJQ","created_at":"2026-07-05T00:52:27Z"},{"alias_kind":"pith_short_8","alias_value":"ZGM5JPI2","created_at":"2026-07-05T00:52:27Z"}],"graph_snapshots":[{"event_id":"sha256:5037cf7c6d009dfb4f67fd33d6e7b0fa643f4b5e89589ecb8fa3fbb5be668496","target":"graph","created_at":"2026-07-05T00:52:27Z","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/1910.01671/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"The critical locus of the loss function of a neural network is determined by the geometry of the functional space and by the parameterization of this space by the network's weights. We introduce a natural distinction between pure critical points, which only depend on the functional space, and spurious critical points, which arise from the parameterization. We apply this perspective to revisit and extend the literature on the loss function of linear neural networks. For this type of network, the functional space is either the set of all linear maps from input to output space, or a determinantal","authors_text":"Joan Bruna, Kathl\\'en Kohn, Matthew Trager","cross_cats":["math.AG","stat.ML"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2019-10-03T18:22:30Z","title":"Pure and Spurious Critical Points: a Geometric Study of Linear Networks"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1910.01671","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:af0817e6f2feaaa458602128c5d07f8145922d9c8ac6a792cf74280b6f5e3633","target":"record","created_at":"2026-07-05T00:52:27Z","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":"0bde633c669432bf4a29b4d650dfe008f34f855fa4594c66e29586c54a1a6c8a","cross_cats_sorted":["math.AG","stat.ML"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2019-10-03T18:22:30Z","title_canon_sha256":"f803ab99b86b7630266c9e406503ab59af0e6840809bebe63c2b76fad4fc7376"},"schema_version":"1.0","source":{"id":"1910.01671","kind":"arxiv","version":2}},"canonical_sha256":"c999d4bd1a148dc79d30db08977996a2fd819630d53a6301c8c80cc76f0e951f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c999d4bd1a148dc79d30db08977996a2fd819630d53a6301c8c80cc76f0e951f","first_computed_at":"2026-07-05T00:52:27.478965Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T00:52:27.478965Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"A/sMQ2YU9hjYszWX5QcGbCcGmQoEMyDwVN5lLSmQa22aaNYfjThnpLFJTF7WjeEQxIKueSxFKR63XNfdRMmQCA==","signature_status":"signed_v1","signed_at":"2026-07-05T00:52:27.479396Z","signed_message":"canonical_sha256_bytes"},"source_id":"1910.01671","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:af0817e6f2feaaa458602128c5d07f8145922d9c8ac6a792cf74280b6f5e3633","sha256:5037cf7c6d009dfb4f67fd33d6e7b0fa643f4b5e89589ecb8fa3fbb5be668496"],"state_sha256":"3c56b82f2cded11da83dd7033cabfbafa1e824e01a6271e450a3a034c3f95e22"}