{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:6ZTQJG7KSNFFLY3TDZ4AMSMKW5","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":"4b46f3bba0ef873ae0a15819f17d45d475fc9b5a5927917a656a91756d77d5bc","cross_cats_sorted":["math.OC","stat.ML"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2019-06-21T16:46:04Z","title_canon_sha256":"61d645102792e881594b04fdc11e0dcc33014ded13e658811775052ccc93f9a0"},"schema_version":"1.0","source":{"id":"1906.09235","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1906.09235","created_at":"2026-05-17T23:41:41Z"},{"alias_kind":"arxiv_version","alias_value":"1906.09235v2","created_at":"2026-05-17T23:41:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1906.09235","created_at":"2026-05-17T23:41:41Z"},{"alias_kind":"pith_short_12","alias_value":"6ZTQJG7KSNFF","created_at":"2026-05-18T12:33:10Z"},{"alias_kind":"pith_short_16","alias_value":"6ZTQJG7KSNFFLY3T","created_at":"2026-05-18T12:33:10Z"},{"alias_kind":"pith_short_8","alias_value":"6ZTQJG7K","created_at":"2026-05-18T12:33:10Z"}],"graph_snapshots":[{"event_id":"sha256:6fcd2e8b164825755d021caf1bd007da485f484baa2f09e73ad0f4bb18f4e0a2","target":"graph","created_at":"2026-05-17T23:41:41Z","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"},"paper":{"abstract_excerpt":"Along with fruitful applications of Deep Neural Networks (DNNs) to realistic problems, recently, some empirical studies of DNNs reported a universal phenomenon of Frequency Principle (F-Principle): a DNN tends to learn a target function from low to high frequencies during the training. The F-Principle has been very useful in providing both qualitative and quantitative understandings of DNNs. In this paper, we rigorously investigate the F-Principle for the training dynamics of a general DNN at three stages: initial stage, intermediate stage, and final stage. For each stage, a theorem is provide","authors_text":"Tao Luo, Yaoyu Zhang, Zheng Ma, Zhi-Qin John Xu","cross_cats":["math.OC","stat.ML"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2019-06-21T16:46:04Z","title":"Theory of the Frequency Principle for General Deep Neural Networks"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1906.09235","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":"sha256:2fc05f6744af30c0dbd0848f9dfe23f84791b2be71fc0aa9727b9dd964662d05","target":"record","created_at":"2026-05-17T23:41:41Z","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":"4b46f3bba0ef873ae0a15819f17d45d475fc9b5a5927917a656a91756d77d5bc","cross_cats_sorted":["math.OC","stat.ML"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2019-06-21T16:46:04Z","title_canon_sha256":"61d645102792e881594b04fdc11e0dcc33014ded13e658811775052ccc93f9a0"},"schema_version":"1.0","source":{"id":"1906.09235","kind":"arxiv","version":2}},"canonical_sha256":"f667049bea934a55e3731e7806498ab75fe689b65cd1b7a0f2471cf23eb43bfd","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f667049bea934a55e3731e7806498ab75fe689b65cd1b7a0f2471cf23eb43bfd","first_computed_at":"2026-05-17T23:41:41.725764Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:41:41.725764Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"47s++KDmTT6OrugUIeUEHcrSRdAjh6Nn6r7X85/ajaZytXotry9Vm7emibJi+IraKEmiFB7Gu9Hp2d4/RSAgDA==","signature_status":"signed_v1","signed_at":"2026-05-17T23:41:41.726238Z","signed_message":"canonical_sha256_bytes"},"source_id":"1906.09235","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2fc05f6744af30c0dbd0848f9dfe23f84791b2be71fc0aa9727b9dd964662d05","sha256:6fcd2e8b164825755d021caf1bd007da485f484baa2f09e73ad0f4bb18f4e0a2"],"state_sha256":"ea7effef900d2a685c8bc3aae25f7a9f86e24a997aca88bf54015e1ffd0d0bcd"}