{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:XCKNJJN26SYX6UOMYANIQT4AG3","short_pith_number":"pith:XCKNJJN2","canonical_record":{"source":{"id":"2402.10127","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"stat.ML","submitted_at":"2024-02-15T17:31:19Z","cross_cats_sorted":["cs.LG","math.PR","math.ST","stat.TH"],"title_canon_sha256":"6c1ad7dd47cdf93197731714778f4717bf8521e82ca41b19428eeec70acb35f0","abstract_canon_sha256":"109137d538d3067e2c5051f674b4b051995aff6d8a46c4552c46ccbb74502869"},"schema_version":"1.0"},"canonical_sha256":"b894d4a5baf4b17f51ccc01a884f8036c330bf9ef392ad065caec2c3b78920cd","source":{"kind":"arxiv","id":"2402.10127","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2402.10127","created_at":"2026-07-05T07:45:41Z"},{"alias_kind":"arxiv_version","alias_value":"2402.10127v1","created_at":"2026-07-05T07:45:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2402.10127","created_at":"2026-07-05T07:45:41Z"},{"alias_kind":"pith_short_12","alias_value":"XCKNJJN26SYX","created_at":"2026-07-05T07:45:41Z"},{"alias_kind":"pith_short_16","alias_value":"XCKNJJN26SYX6UOM","created_at":"2026-07-05T07:45:41Z"},{"alias_kind":"pith_short_8","alias_value":"XCKNJJN2","created_at":"2026-07-05T07:45:41Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:XCKNJJN26SYX6UOMYANIQT4AG3","target":"record","payload":{"canonical_record":{"source":{"id":"2402.10127","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"stat.ML","submitted_at":"2024-02-15T17:31:19Z","cross_cats_sorted":["cs.LG","math.PR","math.ST","stat.TH"],"title_canon_sha256":"6c1ad7dd47cdf93197731714778f4717bf8521e82ca41b19428eeec70acb35f0","abstract_canon_sha256":"109137d538d3067e2c5051f674b4b051995aff6d8a46c4552c46ccbb74502869"},"schema_version":"1.0"},"canonical_sha256":"b894d4a5baf4b17f51ccc01a884f8036c330bf9ef392ad065caec2c3b78920cd","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T07:45:41.678063Z","signature_b64":"GVNqFFHHoc/ZqBWJMr41vBphD3UdQwN/qv2+GAruTC9ZjZtVoHnO8gWnHDWYnGceycRVtuBqXOhSXUhzpe6KDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b894d4a5baf4b17f51ccc01a884f8036c330bf9ef392ad065caec2c3b78920cd","last_reissued_at":"2026-07-05T07:45:41.677636Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T07:45:41.677636Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2402.10127","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":"2026-07-05T07:45:41Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"M7tLUbRW7m0O7O2Zi0KylLrilE3+Ec+AEnJlNUZKdEI7af1cEYaKM+b32dD7RhVMBvblpy2wQ9W3Z0cUxoVCBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-21T22:27:50.050514Z"},"content_sha256":"88a297194d168024bbf274566f432cfb3874899afb0aa7866ebe54cb9d2fe7ad","schema_version":"1.0","event_id":"sha256:88a297194d168024bbf274566f432cfb3874899afb0aa7866ebe54cb9d2fe7ad"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:XCKNJJN26SYX6UOMYANIQT4AG3","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Nonlinear spiked covariance matrices and signal propagation in deep neural networks","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.LG","math.PR","math.ST","stat.TH"],"primary_cat":"stat.ML","authors_text":"Denny Wu, Zhichao Wang, Zhou Fan","submitted_at":"2024-02-15T17:31:19Z","abstract_excerpt":"Many recent works have studied the eigenvalue spectrum of the Conjugate Kernel (CK) defined by the nonlinear feature map of a feedforward neural network. However, existing results only establish weak convergence of the empirical eigenvalue distribution, and fall short of providing precise quantitative characterizations of the ''spike'' eigenvalues and eigenvectors that often capture the low-dimensional signal structure of the learning problem. In this work, we characterize these signal eigenvalues and eigenvectors for a nonlinear version of the spiked covariance model, including the CK as a sp"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2402.10127","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/2402.10127/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":"2026-07-05T07:45:41Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"XKrq+z/Y4yG3KzSshtr9AYvPXfFkL2JN0nfvGm5kHkVNoHKMVLlQs+kSRAxtDVe1DMdozga4FG1BODFli8pZDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-21T22:27:50.051120Z"},"content_sha256":"9e82083e4d13edcfb36c1ce5683b6f2e0e36ee7756d748084d0d6b1292e9b5ef","schema_version":"1.0","event_id":"sha256:9e82083e4d13edcfb36c1ce5683b6f2e0e36ee7756d748084d0d6b1292e9b5ef"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/XCKNJJN26SYX6UOMYANIQT4AG3/bundle.json","state_url":"https://pith.science/pith/XCKNJJN26SYX6UOMYANIQT4AG3/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/XCKNJJN26SYX6UOMYANIQT4AG3/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":"2026-08-21T22:27:50Z","links":{"resolver":"https://pith.science/pith/XCKNJJN26SYX6UOMYANIQT4AG3","bundle":"https://pith.science/pith/XCKNJJN26SYX6UOMYANIQT4AG3/bundle.json","state":"https://pith.science/pith/XCKNJJN26SYX6UOMYANIQT4AG3/state.json","well_known_bundle":"https://pith.science/.well-known/pith/XCKNJJN26SYX6UOMYANIQT4AG3/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:XCKNJJN26SYX6UOMYANIQT4AG3","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":"109137d538d3067e2c5051f674b4b051995aff6d8a46c4552c46ccbb74502869","cross_cats_sorted":["cs.LG","math.PR","math.ST","stat.TH"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"stat.ML","submitted_at":"2024-02-15T17:31:19Z","title_canon_sha256":"6c1ad7dd47cdf93197731714778f4717bf8521e82ca41b19428eeec70acb35f0"},"schema_version":"1.0","source":{"id":"2402.10127","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2402.10127","created_at":"2026-07-05T07:45:41Z"},{"alias_kind":"arxiv_version","alias_value":"2402.10127v1","created_at":"2026-07-05T07:45:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2402.10127","created_at":"2026-07-05T07:45:41Z"},{"alias_kind":"pith_short_12","alias_value":"XCKNJJN26SYX","created_at":"2026-07-05T07:45:41Z"},{"alias_kind":"pith_short_16","alias_value":"XCKNJJN26SYX6UOM","created_at":"2026-07-05T07:45:41Z"},{"alias_kind":"pith_short_8","alias_value":"XCKNJJN2","created_at":"2026-07-05T07:45:41Z"}],"graph_snapshots":[{"event_id":"sha256:9e82083e4d13edcfb36c1ce5683b6f2e0e36ee7756d748084d0d6b1292e9b5ef","target":"graph","created_at":"2026-07-05T07:45:41Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2402.10127/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Many recent works have studied the eigenvalue spectrum of the Conjugate Kernel (CK) defined by the nonlinear feature map of a feedforward neural network. However, existing results only establish weak convergence of the empirical eigenvalue distribution, and fall short of providing precise quantitative characterizations of the ''spike'' eigenvalues and eigenvectors that often capture the low-dimensional signal structure of the learning problem. In this work, we characterize these signal eigenvalues and eigenvectors for a nonlinear version of the spiked covariance model, including the CK as a sp","authors_text":"Denny Wu, Zhichao Wang, Zhou Fan","cross_cats":["cs.LG","math.PR","math.ST","stat.TH"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"stat.ML","submitted_at":"2024-02-15T17:31:19Z","title":"Nonlinear spiked covariance matrices and signal propagation in deep neural networks"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2402.10127","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":"sha256:88a297194d168024bbf274566f432cfb3874899afb0aa7866ebe54cb9d2fe7ad","target":"record","created_at":"2026-07-05T07:45:41Z","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":"109137d538d3067e2c5051f674b4b051995aff6d8a46c4552c46ccbb74502869","cross_cats_sorted":["cs.LG","math.PR","math.ST","stat.TH"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"stat.ML","submitted_at":"2024-02-15T17:31:19Z","title_canon_sha256":"6c1ad7dd47cdf93197731714778f4717bf8521e82ca41b19428eeec70acb35f0"},"schema_version":"1.0","source":{"id":"2402.10127","kind":"arxiv","version":1}},"canonical_sha256":"b894d4a5baf4b17f51ccc01a884f8036c330bf9ef392ad065caec2c3b78920cd","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b894d4a5baf4b17f51ccc01a884f8036c330bf9ef392ad065caec2c3b78920cd","first_computed_at":"2026-07-05T07:45:41.677636Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T07:45:41.677636Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"GVNqFFHHoc/ZqBWJMr41vBphD3UdQwN/qv2+GAruTC9ZjZtVoHnO8gWnHDWYnGceycRVtuBqXOhSXUhzpe6KDg==","signature_status":"signed_v1","signed_at":"2026-07-05T07:45:41.678063Z","signed_message":"canonical_sha256_bytes"},"source_id":"2402.10127","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:88a297194d168024bbf274566f432cfb3874899afb0aa7866ebe54cb9d2fe7ad","sha256:9e82083e4d13edcfb36c1ce5683b6f2e0e36ee7756d748084d0d6b1292e9b5ef"],"state_sha256":"2a2c88c9ea4f464d18487dd089159f2d7f8088ac541ad48e8e77f7d19efa3b9f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GQe1TR9D1FhH4jewUBJIHHS0K6vXIW6iu5RT/vlwTDWeOknTzyJpUMV3n9X0KP3XvKvNw6rGZaCfB8uG7a0uDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-21T22:27:50.065574Z","bundle_sha256":"f06f042590ee1b94e32649c9f1d7ecfc2f6e85a8e4621ae24666dfe52b192891"}}