{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2021:XXJBPSGG25634V6POF3XTSV36M","short_pith_number":"pith:XXJBPSGG","canonical_record":{"source":{"id":"2108.02040","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2021-08-04T13:10:30Z","cross_cats_sorted":["math.OC"],"title_canon_sha256":"413165efe0e78318f1cb842bbad1f30314c322788b571f262ec4ef2a08d78152","abstract_canon_sha256":"2e11a8c68a9cb40a804cd0628c239c003dfcbf9ee36144a81b40bf717c0debaf"},"schema_version":"1.0"},"canonical_sha256":"bdd217c8c6d77dbe57cf717779cabbf3350b7af09a4c2e6692ad1dd1321c8c97","source":{"kind":"arxiv","id":"2108.02040","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2108.02040","created_at":"2026-07-05T03:34:45Z"},{"alias_kind":"arxiv_version","alias_value":"2108.02040v2","created_at":"2026-07-05T03:34:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2108.02040","created_at":"2026-07-05T03:34:45Z"},{"alias_kind":"pith_short_12","alias_value":"XXJBPSGG2563","created_at":"2026-07-05T03:34:45Z"},{"alias_kind":"pith_short_16","alias_value":"XXJBPSGG25634V6P","created_at":"2026-07-05T03:34:45Z"},{"alias_kind":"pith_short_8","alias_value":"XXJBPSGG","created_at":"2026-07-05T03:34:45Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2021:XXJBPSGG25634V6POF3XTSV36M","target":"record","payload":{"canonical_record":{"source":{"id":"2108.02040","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2021-08-04T13:10:30Z","cross_cats_sorted":["math.OC"],"title_canon_sha256":"413165efe0e78318f1cb842bbad1f30314c322788b571f262ec4ef2a08d78152","abstract_canon_sha256":"2e11a8c68a9cb40a804cd0628c239c003dfcbf9ee36144a81b40bf717c0debaf"},"schema_version":"1.0"},"canonical_sha256":"bdd217c8c6d77dbe57cf717779cabbf3350b7af09a4c2e6692ad1dd1321c8c97","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T03:34:45.924080Z","signature_b64":"8e+lE3WQ+Phwo2fGzpcgRw0k2zrmcGz3Hc8FmUyM8S3FVBlDfryeleLUudnm3EAFYvSlpFLOHqOYf0k2ZyzXAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bdd217c8c6d77dbe57cf717779cabbf3350b7af09a4c2e6692ad1dd1321c8c97","last_reissued_at":"2026-07-05T03:34:45.923659Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T03:34:45.923659Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2108.02040","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T03:34:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4U+hbEHPyi5kKfFtFobCzz4HVg9QUT5sHe7DEBjWRTRm/yH57UvA/9iaz/gBQj6DmkleMbbCsGdWkWPkfA71CA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-17T20:24:22.475760Z"},"content_sha256":"4534c85d84b04f4e69e6105fa4981db2783d3dbd4a56e4f265ad0203ba5c0fab","schema_version":"1.0","event_id":"sha256:4534c85d84b04f4e69e6105fa4981db2783d3dbd4a56e4f265ad0203ba5c0fab"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2021:XXJBPSGG25634V6POF3XTSV36M","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Convergence of gradient descent for learning linear neural networks","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.OC"],"primary_cat":"cs.LG","authors_text":"Gabin Maxime Nguegnang, Holger Rauhut, Ulrich Terstiege","submitted_at":"2021-08-04T13:10:30Z","abstract_excerpt":"We study the convergence properties of gradient descent for training deep linear neural networks, i.e., deep matrix factorizations, by extending a previous analysis for the related gradient flow. We show that under suitable conditions on the step sizes gradient descent converges to a critical point of the loss function, i.e., the square loss in this article. Furthermore, we demonstrate that for almost all initializations gradient descent converges to a global minimum in the case of two layers. In the case of three or more layers we show that gradient descent converges to a global minimum on th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2108.02040","kind":"arxiv","version":2},"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/2108.02040/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":"2026-07-05T03:34:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"De22g+8xg+NblTqIO02bhrbSJXU+S/jSMSjf+MWmj/EoYTE14uQmkolPa2aHTDW1yyfIHOj0X1MPtOzUTYEkCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-17T20:24:22.476137Z"},"content_sha256":"9e13829b79dd27becfb3c65f9bf36b4b36a5cfb5ae93fe75efd3d2f9d6f53439","schema_version":"1.0","event_id":"sha256:9e13829b79dd27becfb3c65f9bf36b4b36a5cfb5ae93fe75efd3d2f9d6f53439"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/XXJBPSGG25634V6POF3XTSV36M/bundle.json","state_url":"https://pith.science/pith/XXJBPSGG25634V6POF3XTSV36M/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/XXJBPSGG25634V6POF3XTSV36M/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":"2026-08-17T20:24:22Z","links":{"resolver":"https://pith.science/pith/XXJBPSGG25634V6POF3XTSV36M","bundle":"https://pith.science/pith/XXJBPSGG25634V6POF3XTSV36M/bundle.json","state":"https://pith.science/pith/XXJBPSGG25634V6POF3XTSV36M/state.json","well_known_bundle":"https://pith.science/.well-known/pith/XXJBPSGG25634V6POF3XTSV36M/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2021:XXJBPSGG25634V6POF3XTSV36M","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":"2e11a8c68a9cb40a804cd0628c239c003dfcbf9ee36144a81b40bf717c0debaf","cross_cats_sorted":["math.OC"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2021-08-04T13:10:30Z","title_canon_sha256":"413165efe0e78318f1cb842bbad1f30314c322788b571f262ec4ef2a08d78152"},"schema_version":"1.0","source":{"id":"2108.02040","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2108.02040","created_at":"2026-07-05T03:34:45Z"},{"alias_kind":"arxiv_version","alias_value":"2108.02040v2","created_at":"2026-07-05T03:34:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2108.02040","created_at":"2026-07-05T03:34:45Z"},{"alias_kind":"pith_short_12","alias_value":"XXJBPSGG2563","created_at":"2026-07-05T03:34:45Z"},{"alias_kind":"pith_short_16","alias_value":"XXJBPSGG25634V6P","created_at":"2026-07-05T03:34:45Z"},{"alias_kind":"pith_short_8","alias_value":"XXJBPSGG","created_at":"2026-07-05T03:34:45Z"}],"graph_snapshots":[{"event_id":"sha256:9e13829b79dd27becfb3c65f9bf36b4b36a5cfb5ae93fe75efd3d2f9d6f53439","target":"graph","created_at":"2026-07-05T03:34:45Z","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/2108.02040/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We study the convergence properties of gradient descent for training deep linear neural networks, i.e., deep matrix factorizations, by extending a previous analysis for the related gradient flow. We show that under suitable conditions on the step sizes gradient descent converges to a critical point of the loss function, i.e., the square loss in this article. Furthermore, we demonstrate that for almost all initializations gradient descent converges to a global minimum in the case of two layers. In the case of three or more layers we show that gradient descent converges to a global minimum on th","authors_text":"Gabin Maxime Nguegnang, Holger Rauhut, Ulrich Terstiege","cross_cats":["math.OC"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2021-08-04T13:10:30Z","title":"Convergence of gradient descent for learning linear neural networks"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2108.02040","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":"sha256:4534c85d84b04f4e69e6105fa4981db2783d3dbd4a56e4f265ad0203ba5c0fab","target":"record","created_at":"2026-07-05T03:34:45Z","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":"2e11a8c68a9cb40a804cd0628c239c003dfcbf9ee36144a81b40bf717c0debaf","cross_cats_sorted":["math.OC"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2021-08-04T13:10:30Z","title_canon_sha256":"413165efe0e78318f1cb842bbad1f30314c322788b571f262ec4ef2a08d78152"},"schema_version":"1.0","source":{"id":"2108.02040","kind":"arxiv","version":2}},"canonical_sha256":"bdd217c8c6d77dbe57cf717779cabbf3350b7af09a4c2e6692ad1dd1321c8c97","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"bdd217c8c6d77dbe57cf717779cabbf3350b7af09a4c2e6692ad1dd1321c8c97","first_computed_at":"2026-07-05T03:34:45.923659Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T03:34:45.923659Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"8e+lE3WQ+Phwo2fGzpcgRw0k2zrmcGz3Hc8FmUyM8S3FVBlDfryeleLUudnm3EAFYvSlpFLOHqOYf0k2ZyzXAA==","signature_status":"signed_v1","signed_at":"2026-07-05T03:34:45.924080Z","signed_message":"canonical_sha256_bytes"},"source_id":"2108.02040","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4534c85d84b04f4e69e6105fa4981db2783d3dbd4a56e4f265ad0203ba5c0fab","sha256:9e13829b79dd27becfb3c65f9bf36b4b36a5cfb5ae93fe75efd3d2f9d6f53439"],"state_sha256":"b47c3861f1cad5dc34573ea18e47d93e185bdf13b87c7de8fd295103e801394e"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"SrxsXviX1otKr4HhbEWJ4LRmX/1NSj6YDesMIdtQbRlIjuEADaNASclc4d1tqrb5NeXP7H/7krIBOP9me4zFBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-17T20:24:22.478496Z","bundle_sha256":"70d188e8c11f12781fd077d07a8bfacbf9965e00ba70b818ed12d0ee1d42ad68"}}