{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2021:ER5HVS7KRT34Q4TVW5KRNB6EUP","short_pith_number":"pith:ER5HVS7K","canonical_record":{"source":{"id":"2102.12430","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2021-02-24T17:50:17Z","cross_cats_sorted":["stat.ML"],"title_canon_sha256":"9ac2f5ec0d05cbe5c3e112da7fadad9b564888a6962d577e76fb98ca5f0e0123","abstract_canon_sha256":"2369b8b8ae4d1ec861e2e72cee1e876eecd518f1db1f46dc4b095f40db279d99"},"schema_version":"1.0"},"canonical_sha256":"247a7acbea8cf7c87275b7551687c4a3c9ed92479b8122edb268d1a0ee7219f4","source":{"kind":"arxiv","id":"2102.12430","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2102.12430","created_at":"2026-07-05T02:18:12Z"},{"alias_kind":"arxiv_version","alias_value":"2102.12430v1","created_at":"2026-07-05T02:18:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2102.12430","created_at":"2026-07-05T02:18:12Z"},{"alias_kind":"pith_short_12","alias_value":"ER5HVS7KRT34","created_at":"2026-07-05T02:18:12Z"},{"alias_kind":"pith_short_16","alias_value":"ER5HVS7KRT34Q4TV","created_at":"2026-07-05T02:18:12Z"},{"alias_kind":"pith_short_8","alias_value":"ER5HVS7K","created_at":"2026-07-05T02:18:12Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2021:ER5HVS7KRT34Q4TVW5KRNB6EUP","target":"record","payload":{"canonical_record":{"source":{"id":"2102.12430","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2021-02-24T17:50:17Z","cross_cats_sorted":["stat.ML"],"title_canon_sha256":"9ac2f5ec0d05cbe5c3e112da7fadad9b564888a6962d577e76fb98ca5f0e0123","abstract_canon_sha256":"2369b8b8ae4d1ec861e2e72cee1e876eecd518f1db1f46dc4b095f40db279d99"},"schema_version":"1.0"},"canonical_sha256":"247a7acbea8cf7c87275b7551687c4a3c9ed92479b8122edb268d1a0ee7219f4","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T02:18:12.763219Z","signature_b64":"lh00z5u7eQADKscE9uNrMXgYgMPGNemOgztbqbtIp4G1eJ7k5TMB8dXQKMhYoSTFfGaffJyaQEKvwy7IVHXhDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"247a7acbea8cf7c87275b7551687c4a3c9ed92479b8122edb268d1a0ee7219f4","last_reissued_at":"2026-07-05T02:18:12.762764Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T02:18:12.762764Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2102.12430","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T02:18:12Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"gTwd1BDWhwgJ4WKFbVPG5piuMyuzts5n+TfhFTr9NFz86JTfW61rotvT4aZqzNq9GLJjwiynRPXcgtbQOUNrAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-20T15:30:08.715457Z"},"content_sha256":"6a00fc9b92209d2aa7f0a6dba71ce1e713912a65fb46df46d0f00efeb14aad92","schema_version":"1.0","event_id":"sha256:6a00fc9b92209d2aa7f0a6dba71ce1e713912a65fb46df46d0f00efeb14aad92"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2021:ER5HVS7KRT34Q4TVW5KRNB6EUP","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Noisy Gradient Descent Converges to Flat Minima for Nonconvex Matrix Factorization","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["stat.ML"],"primary_cat":"cs.LG","authors_text":"Enlu Zhou, Song Wei, Tianyi Liu, Tuo Zhao, Yan Li","submitted_at":"2021-02-24T17:50:17Z","abstract_excerpt":"Numerous empirical evidences have corroborated the importance of noise in nonconvex optimization problems. The theory behind such empirical observations, however, is still largely unknown. This paper studies this fundamental problem through investigating the nonconvex rectangular matrix factorization problem, which has infinitely many global minima due to rotation and scaling invariance. Hence, gradient descent (GD) can converge to any optimum, depending on the initialization. In contrast, we show that a perturbed form of GD with an arbitrary initialization converges to a global optimum that i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2102.12430","kind":"arxiv","version":1},"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/2102.12430/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":"2026-07-05T02:18:12Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ujKM454eSxq1ETFKapeXgbqBmsAxBi0YlfP2nmESeCWWy79qqZ381xOloOOJ2hMvdETmIibOtKc5lN9O/wrMCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-20T15:30:08.716020Z"},"content_sha256":"563c30f7b53596a48fccd9368c26a905356093501cc1bb83da0fa627423ba7ae","schema_version":"1.0","event_id":"sha256:563c30f7b53596a48fccd9368c26a905356093501cc1bb83da0fa627423ba7ae"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ER5HVS7KRT34Q4TVW5KRNB6EUP/bundle.json","state_url":"https://pith.science/pith/ER5HVS7KRT34Q4TVW5KRNB6EUP/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ER5HVS7KRT34Q4TVW5KRNB6EUP/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":"2026-08-20T15:30:08Z","links":{"resolver":"https://pith.science/pith/ER5HVS7KRT34Q4TVW5KRNB6EUP","bundle":"https://pith.science/pith/ER5HVS7KRT34Q4TVW5KRNB6EUP/bundle.json","state":"https://pith.science/pith/ER5HVS7KRT34Q4TVW5KRNB6EUP/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ER5HVS7KRT34Q4TVW5KRNB6EUP/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2021:ER5HVS7KRT34Q4TVW5KRNB6EUP","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":"2369b8b8ae4d1ec861e2e72cee1e876eecd518f1db1f46dc4b095f40db279d99","cross_cats_sorted":["stat.ML"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2021-02-24T17:50:17Z","title_canon_sha256":"9ac2f5ec0d05cbe5c3e112da7fadad9b564888a6962d577e76fb98ca5f0e0123"},"schema_version":"1.0","source":{"id":"2102.12430","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2102.12430","created_at":"2026-07-05T02:18:12Z"},{"alias_kind":"arxiv_version","alias_value":"2102.12430v1","created_at":"2026-07-05T02:18:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2102.12430","created_at":"2026-07-05T02:18:12Z"},{"alias_kind":"pith_short_12","alias_value":"ER5HVS7KRT34","created_at":"2026-07-05T02:18:12Z"},{"alias_kind":"pith_short_16","alias_value":"ER5HVS7KRT34Q4TV","created_at":"2026-07-05T02:18:12Z"},{"alias_kind":"pith_short_8","alias_value":"ER5HVS7K","created_at":"2026-07-05T02:18:12Z"}],"graph_snapshots":[{"event_id":"sha256:563c30f7b53596a48fccd9368c26a905356093501cc1bb83da0fa627423ba7ae","target":"graph","created_at":"2026-07-05T02:18:12Z","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/2102.12430/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Numerous empirical evidences have corroborated the importance of noise in nonconvex optimization problems. The theory behind such empirical observations, however, is still largely unknown. This paper studies this fundamental problem through investigating the nonconvex rectangular matrix factorization problem, which has infinitely many global minima due to rotation and scaling invariance. Hence, gradient descent (GD) can converge to any optimum, depending on the initialization. In contrast, we show that a perturbed form of GD with an arbitrary initialization converges to a global optimum that i","authors_text":"Enlu Zhou, Song Wei, Tianyi Liu, Tuo Zhao, Yan Li","cross_cats":["stat.ML"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2021-02-24T17:50:17Z","title":"Noisy Gradient Descent Converges to Flat Minima for Nonconvex Matrix Factorization"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2102.12430","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":"sha256:6a00fc9b92209d2aa7f0a6dba71ce1e713912a65fb46df46d0f00efeb14aad92","target":"record","created_at":"2026-07-05T02:18:12Z","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":"2369b8b8ae4d1ec861e2e72cee1e876eecd518f1db1f46dc4b095f40db279d99","cross_cats_sorted":["stat.ML"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2021-02-24T17:50:17Z","title_canon_sha256":"9ac2f5ec0d05cbe5c3e112da7fadad9b564888a6962d577e76fb98ca5f0e0123"},"schema_version":"1.0","source":{"id":"2102.12430","kind":"arxiv","version":1}},"canonical_sha256":"247a7acbea8cf7c87275b7551687c4a3c9ed92479b8122edb268d1a0ee7219f4","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"247a7acbea8cf7c87275b7551687c4a3c9ed92479b8122edb268d1a0ee7219f4","first_computed_at":"2026-07-05T02:18:12.762764Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T02:18:12.762764Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"lh00z5u7eQADKscE9uNrMXgYgMPGNemOgztbqbtIp4G1eJ7k5TMB8dXQKMhYoSTFfGaffJyaQEKvwy7IVHXhDA==","signature_status":"signed_v1","signed_at":"2026-07-05T02:18:12.763219Z","signed_message":"canonical_sha256_bytes"},"source_id":"2102.12430","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6a00fc9b92209d2aa7f0a6dba71ce1e713912a65fb46df46d0f00efeb14aad92","sha256:563c30f7b53596a48fccd9368c26a905356093501cc1bb83da0fa627423ba7ae"],"state_sha256":"6cde241f712f84f270943d448b829eb198e58532c15a1678062ae60365442c31"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Wp28L/qeDPc/nyAwyyNGHUcu5PUK0GbYNS4mASGaNjp+EaV9zG7VZ7iLvNoayECk6YY+8OWQMKZ12yqkqZftAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-20T15:30:08.721926Z","bundle_sha256":"74e70769cbfced7989f86ac75cebe4b33d9c3365227fef8670ade399baa7e04d"}}