{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:SFEWRGXROK4W2MZJV5PJVGAECU","short_pith_number":"pith:SFEWRGXR","canonical_record":{"source":{"id":"1605.07110","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"stat.ML","submitted_at":"2016-05-23T17:34:20Z","cross_cats_sorted":["cs.LG","math.OC"],"title_canon_sha256":"90b52dbb8e965282700beddf97330fef08a8a09dcd218ac63e3859262a46656a","abstract_canon_sha256":"f5857987c78deab04c8e8b9eaa3a76a77cb3b6e38af3509d0542f8a69be14596"},"schema_version":"1.0"},"canonical_sha256":"9149689af172b96d3329af5e9a98041512c984836726403e42cba15ad16bdbce","source":{"kind":"arxiv","id":"1605.07110","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1605.07110","created_at":"2026-05-18T00:53:50Z"},{"alias_kind":"arxiv_version","alias_value":"1605.07110v3","created_at":"2026-05-18T00:53:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1605.07110","created_at":"2026-05-18T00:53:50Z"},{"alias_kind":"pith_short_12","alias_value":"SFEWRGXROK4W","created_at":"2026-05-18T12:30:44Z"},{"alias_kind":"pith_short_16","alias_value":"SFEWRGXROK4W2MZJ","created_at":"2026-05-18T12:30:44Z"},{"alias_kind":"pith_short_8","alias_value":"SFEWRGXR","created_at":"2026-05-18T12:30:44Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:SFEWRGXROK4W2MZJV5PJVGAECU","target":"record","payload":{"canonical_record":{"source":{"id":"1605.07110","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"stat.ML","submitted_at":"2016-05-23T17:34:20Z","cross_cats_sorted":["cs.LG","math.OC"],"title_canon_sha256":"90b52dbb8e965282700beddf97330fef08a8a09dcd218ac63e3859262a46656a","abstract_canon_sha256":"f5857987c78deab04c8e8b9eaa3a76a77cb3b6e38af3509d0542f8a69be14596"},"schema_version":"1.0"},"canonical_sha256":"9149689af172b96d3329af5e9a98041512c984836726403e42cba15ad16bdbce","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:53:50.931440Z","signature_b64":"m33gqIJZi4HSFCu6nDwEEWOkwCdDoIi3hQ3dJNQG0l3crQnCUcgcbfSysrX7e4IT8sUJ9kRljIIrr4coxF2qAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9149689af172b96d3329af5e9a98041512c984836726403e42cba15ad16bdbce","last_reissued_at":"2026-05-18T00:53:50.930970Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:53:50.930970Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1605.07110","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:53:50Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"kBbdOwHqvVpqeYJcN1Li1tkEDvJa6EIcRPRVIDCYTPHXy09RvZzt2y8Eaz6A4Jpp6tLTrAw4hs6YHZud2m9JBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T09:47:13.980862Z"},"content_sha256":"0ccbc163d76979170a151ab3f77b01a8d1d171d163412b591b5bc1c81a64c427","schema_version":"1.0","event_id":"sha256:0ccbc163d76979170a151ab3f77b01a8d1d171d163412b591b5bc1c81a64c427"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:SFEWRGXROK4W2MZJV5PJVGAECU","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Deep Learning without Poor Local Minima","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.LG","math.OC"],"primary_cat":"stat.ML","authors_text":"Kenji Kawaguchi","submitted_at":"2016-05-23T17:34:20Z","abstract_excerpt":"In this paper, we prove a conjecture published in 1989 and also partially address an open problem announced at the Conference on Learning Theory (COLT) 2015. With no unrealistic assumption, we first prove the following statements for the squared loss function of deep linear neural networks with any depth and any widths: 1) the function is non-convex and non-concave, 2) every local minimum is a global minimum, 3) every critical point that is not a global minimum is a saddle point, and 4) there exist \"bad\" saddle points (where the Hessian has no negative eigenvalue) for the deeper networks (with"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1605.07110","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":""},"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":"2026-05-18T00:53:50Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"gXSdbds1QwjQe7PyyAXeLRqPIc/hqvxrXaIkUj1ZuI4u5Vl9LasHhLj29AvDt3+hpuuN97FoE7uK24JYAi5MCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T09:47:13.981461Z"},"content_sha256":"b3fcd8b8a2e190d18ff8461b9e9c9dd64a4db077378c25217f864c7fe4fdb3c7","schema_version":"1.0","event_id":"sha256:b3fcd8b8a2e190d18ff8461b9e9c9dd64a4db077378c25217f864c7fe4fdb3c7"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/SFEWRGXROK4W2MZJV5PJVGAECU/bundle.json","state_url":"https://pith.science/pith/SFEWRGXROK4W2MZJV5PJVGAECU/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/SFEWRGXROK4W2MZJV5PJVGAECU/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":"2026-06-07T09:47:13Z","links":{"resolver":"https://pith.science/pith/SFEWRGXROK4W2MZJV5PJVGAECU","bundle":"https://pith.science/pith/SFEWRGXROK4W2MZJV5PJVGAECU/bundle.json","state":"https://pith.science/pith/SFEWRGXROK4W2MZJV5PJVGAECU/state.json","well_known_bundle":"https://pith.science/.well-known/pith/SFEWRGXROK4W2MZJV5PJVGAECU/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:SFEWRGXROK4W2MZJV5PJVGAECU","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":"f5857987c78deab04c8e8b9eaa3a76a77cb3b6e38af3509d0542f8a69be14596","cross_cats_sorted":["cs.LG","math.OC"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"stat.ML","submitted_at":"2016-05-23T17:34:20Z","title_canon_sha256":"90b52dbb8e965282700beddf97330fef08a8a09dcd218ac63e3859262a46656a"},"schema_version":"1.0","source":{"id":"1605.07110","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1605.07110","created_at":"2026-05-18T00:53:50Z"},{"alias_kind":"arxiv_version","alias_value":"1605.07110v3","created_at":"2026-05-18T00:53:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1605.07110","created_at":"2026-05-18T00:53:50Z"},{"alias_kind":"pith_short_12","alias_value":"SFEWRGXROK4W","created_at":"2026-05-18T12:30:44Z"},{"alias_kind":"pith_short_16","alias_value":"SFEWRGXROK4W2MZJ","created_at":"2026-05-18T12:30:44Z"},{"alias_kind":"pith_short_8","alias_value":"SFEWRGXR","created_at":"2026-05-18T12:30:44Z"}],"graph_snapshots":[{"event_id":"sha256:b3fcd8b8a2e190d18ff8461b9e9c9dd64a4db077378c25217f864c7fe4fdb3c7","target":"graph","created_at":"2026-05-18T00:53:50Z","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":"In this paper, we prove a conjecture published in 1989 and also partially address an open problem announced at the Conference on Learning Theory (COLT) 2015. With no unrealistic assumption, we first prove the following statements for the squared loss function of deep linear neural networks with any depth and any widths: 1) the function is non-convex and non-concave, 2) every local minimum is a global minimum, 3) every critical point that is not a global minimum is a saddle point, and 4) there exist \"bad\" saddle points (where the Hessian has no negative eigenvalue) for the deeper networks (with","authors_text":"Kenji Kawaguchi","cross_cats":["cs.LG","math.OC"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"stat.ML","submitted_at":"2016-05-23T17:34:20Z","title":"Deep Learning without Poor Local Minima"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1605.07110","kind":"arxiv","version":3},"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":"sha256:0ccbc163d76979170a151ab3f77b01a8d1d171d163412b591b5bc1c81a64c427","target":"record","created_at":"2026-05-18T00:53:50Z","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":"f5857987c78deab04c8e8b9eaa3a76a77cb3b6e38af3509d0542f8a69be14596","cross_cats_sorted":["cs.LG","math.OC"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"stat.ML","submitted_at":"2016-05-23T17:34:20Z","title_canon_sha256":"90b52dbb8e965282700beddf97330fef08a8a09dcd218ac63e3859262a46656a"},"schema_version":"1.0","source":{"id":"1605.07110","kind":"arxiv","version":3}},"canonical_sha256":"9149689af172b96d3329af5e9a98041512c984836726403e42cba15ad16bdbce","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9149689af172b96d3329af5e9a98041512c984836726403e42cba15ad16bdbce","first_computed_at":"2026-05-18T00:53:50.930970Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:53:50.930970Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"m33gqIJZi4HSFCu6nDwEEWOkwCdDoIi3hQ3dJNQG0l3crQnCUcgcbfSysrX7e4IT8sUJ9kRljIIrr4coxF2qAg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:53:50.931440Z","signed_message":"canonical_sha256_bytes"},"source_id":"1605.07110","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0ccbc163d76979170a151ab3f77b01a8d1d171d163412b591b5bc1c81a64c427","sha256:b3fcd8b8a2e190d18ff8461b9e9c9dd64a4db077378c25217f864c7fe4fdb3c7"],"state_sha256":"c8dbb75bc139dd7da40893e00a238c42c1dc245971dac7af24e052a25ba00082"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"oIa9oyrhOOU3e5YYtaIjDuk+i2LJR11+G4n+zglL+OGLbIja40Jr1Vg1X2xvrTPOOtvgbJfh+s4idWeaOY6mCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-07T09:47:13.984606Z","bundle_sha256":"b4c48352fd6e1b72f178e407179e4355b955ac09ca1cd9fc5ef89bd5edf5b4da"}}