{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2022:PK3HZ3QF6EC765ZXKY2SQGRBI4","short_pith_number":"pith:PK3HZ3QF","canonical_record":{"source":{"id":"2209.07736","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2022-09-16T06:36:06Z","cross_cats_sorted":["cs.AI"],"title_canon_sha256":"f8d6e5545bd2f65e03908d3f2deec0d887e13d538f99f04153fbee8c3dcf66b9","abstract_canon_sha256":"0c1bbce51fc672ad23ceef518da4bc17ef63901b803dfe471d8d57779b06cfae"},"schema_version":"1.0"},"canonical_sha256":"7ab67cee05f105ff77375635281a214730eb80dc00acd0633138e81a4fbe950f","source":{"kind":"arxiv","id":"2209.07736","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2209.07736","created_at":"2026-07-05T05:07:03Z"},{"alias_kind":"arxiv_version","alias_value":"2209.07736v2","created_at":"2026-07-05T05:07:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2209.07736","created_at":"2026-07-05T05:07:03Z"},{"alias_kind":"pith_short_12","alias_value":"PK3HZ3QF6EC7","created_at":"2026-07-05T05:07:03Z"},{"alias_kind":"pith_short_16","alias_value":"PK3HZ3QF6EC765ZX","created_at":"2026-07-05T05:07:03Z"},{"alias_kind":"pith_short_8","alias_value":"PK3HZ3QF","created_at":"2026-07-05T05:07:03Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2022:PK3HZ3QF6EC765ZXKY2SQGRBI4","target":"record","payload":{"canonical_record":{"source":{"id":"2209.07736","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2022-09-16T06:36:06Z","cross_cats_sorted":["cs.AI"],"title_canon_sha256":"f8d6e5545bd2f65e03908d3f2deec0d887e13d538f99f04153fbee8c3dcf66b9","abstract_canon_sha256":"0c1bbce51fc672ad23ceef518da4bc17ef63901b803dfe471d8d57779b06cfae"},"schema_version":"1.0"},"canonical_sha256":"7ab67cee05f105ff77375635281a214730eb80dc00acd0633138e81a4fbe950f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T05:07:03.043549Z","signature_b64":"VbFv9LdRyddI3i04H1RudFIjTHi8gt1QV3LtIaUZn4TPf+Z5Eh0REJpAqPNAKFhwpvA7qGb0Hb0lmr9LDYyJBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7ab67cee05f105ff77375635281a214730eb80dc00acd0633138e81a4fbe950f","last_reissued_at":"2026-07-05T05:07:03.043080Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T05:07:03.043080Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2209.07736","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":"2026-07-05T05:07:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"o4y+71OW5pYzOzQpRyNkr9WR6S47DGDvG6Ev/rm3JOSfgNne/Ag5y458enInCEjeoqiEfzi6zjviWQv4E1XiAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-12T17:07:33.750777Z"},"content_sha256":"f0500f9a50200c96381da2f4fc8cafdf4e368166905366c14d16015df7541486","schema_version":"1.0","event_id":"sha256:f0500f9a50200c96381da2f4fc8cafdf4e368166905366c14d16015df7541486"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2022:PK3HZ3QF6EC765ZXKY2SQGRBI4","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Extrapolation and Spectral Bias of Neural Nets with Hadamard Product: a Polynomial Net Study","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.AI"],"primary_cat":"cs.LG","authors_text":"Fanghui Liu, Grigorios G Chrysos, Volkan Cevher, Yongtao Wu, Zhenyu Zhu","submitted_at":"2022-09-16T06:36:06Z","abstract_excerpt":"Neural tangent kernel (NTK) is a powerful tool to analyze training dynamics of neural networks and their generalization bounds. The study on NTK has been devoted to typical neural network architectures, but it is incomplete for neural networks with Hadamard products (NNs-Hp), e.g., StyleGAN and polynomial neural networks (PNNs). In this work, we derive the finite-width NTK formulation for a special class of NNs-Hp, i.e., polynomial neural networks. We prove their equivalence to the kernel regression predictor with the associated NTK, which expands the application scope of NTK. Based on our res"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2209.07736","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/2209.07736/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":"2026-07-05T05:07:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"e5JGPL2/+3fs1BipJD7JuC1EbrL2A3cFnAqtveuedY9HcYb+IXBAibauImahblJik2EozislO2OqjFd+PAcMCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-12T17:07:33.752275Z"},"content_sha256":"fd5055d6bf5d62505446568de29d3027ba55860bab415f87aaed1324fa70eecd","schema_version":"1.0","event_id":"sha256:fd5055d6bf5d62505446568de29d3027ba55860bab415f87aaed1324fa70eecd"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/PK3HZ3QF6EC765ZXKY2SQGRBI4/bundle.json","state_url":"https://pith.science/pith/PK3HZ3QF6EC765ZXKY2SQGRBI4/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/PK3HZ3QF6EC765ZXKY2SQGRBI4/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":"2026-08-12T17:07:33Z","links":{"resolver":"https://pith.science/pith/PK3HZ3QF6EC765ZXKY2SQGRBI4","bundle":"https://pith.science/pith/PK3HZ3QF6EC765ZXKY2SQGRBI4/bundle.json","state":"https://pith.science/pith/PK3HZ3QF6EC765ZXKY2SQGRBI4/state.json","well_known_bundle":"https://pith.science/.well-known/pith/PK3HZ3QF6EC765ZXKY2SQGRBI4/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2022:PK3HZ3QF6EC765ZXKY2SQGRBI4","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":"0c1bbce51fc672ad23ceef518da4bc17ef63901b803dfe471d8d57779b06cfae","cross_cats_sorted":["cs.AI"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2022-09-16T06:36:06Z","title_canon_sha256":"f8d6e5545bd2f65e03908d3f2deec0d887e13d538f99f04153fbee8c3dcf66b9"},"schema_version":"1.0","source":{"id":"2209.07736","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2209.07736","created_at":"2026-07-05T05:07:03Z"},{"alias_kind":"arxiv_version","alias_value":"2209.07736v2","created_at":"2026-07-05T05:07:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2209.07736","created_at":"2026-07-05T05:07:03Z"},{"alias_kind":"pith_short_12","alias_value":"PK3HZ3QF6EC7","created_at":"2026-07-05T05:07:03Z"},{"alias_kind":"pith_short_16","alias_value":"PK3HZ3QF6EC765ZX","created_at":"2026-07-05T05:07:03Z"},{"alias_kind":"pith_short_8","alias_value":"PK3HZ3QF","created_at":"2026-07-05T05:07:03Z"}],"graph_snapshots":[{"event_id":"sha256:fd5055d6bf5d62505446568de29d3027ba55860bab415f87aaed1324fa70eecd","target":"graph","created_at":"2026-07-05T05:07:03Z","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/2209.07736/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Neural tangent kernel (NTK) is a powerful tool to analyze training dynamics of neural networks and their generalization bounds. The study on NTK has been devoted to typical neural network architectures, but it is incomplete for neural networks with Hadamard products (NNs-Hp), e.g., StyleGAN and polynomial neural networks (PNNs). In this work, we derive the finite-width NTK formulation for a special class of NNs-Hp, i.e., polynomial neural networks. We prove their equivalence to the kernel regression predictor with the associated NTK, which expands the application scope of NTK. Based on our res","authors_text":"Fanghui Liu, Grigorios G Chrysos, Volkan Cevher, Yongtao Wu, Zhenyu Zhu","cross_cats":["cs.AI"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2022-09-16T06:36:06Z","title":"Extrapolation and Spectral Bias of Neural Nets with Hadamard Product: a Polynomial Net Study"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2209.07736","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:f0500f9a50200c96381da2f4fc8cafdf4e368166905366c14d16015df7541486","target":"record","created_at":"2026-07-05T05:07:03Z","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":"0c1bbce51fc672ad23ceef518da4bc17ef63901b803dfe471d8d57779b06cfae","cross_cats_sorted":["cs.AI"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2022-09-16T06:36:06Z","title_canon_sha256":"f8d6e5545bd2f65e03908d3f2deec0d887e13d538f99f04153fbee8c3dcf66b9"},"schema_version":"1.0","source":{"id":"2209.07736","kind":"arxiv","version":2}},"canonical_sha256":"7ab67cee05f105ff77375635281a214730eb80dc00acd0633138e81a4fbe950f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7ab67cee05f105ff77375635281a214730eb80dc00acd0633138e81a4fbe950f","first_computed_at":"2026-07-05T05:07:03.043080Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T05:07:03.043080Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"VbFv9LdRyddI3i04H1RudFIjTHi8gt1QV3LtIaUZn4TPf+Z5Eh0REJpAqPNAKFhwpvA7qGb0Hb0lmr9LDYyJBw==","signature_status":"signed_v1","signed_at":"2026-07-05T05:07:03.043549Z","signed_message":"canonical_sha256_bytes"},"source_id":"2209.07736","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f0500f9a50200c96381da2f4fc8cafdf4e368166905366c14d16015df7541486","sha256:fd5055d6bf5d62505446568de29d3027ba55860bab415f87aaed1324fa70eecd"],"state_sha256":"3942a4c052ada8521accb5a90ab5e7ebf012b693d90f51f7f67e5c7da28347d5"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"pX/Aseriw2j9PXKF0f0lQxbeWS3EBqAj3k6ekz1crktkzkQ37Lc9qRmne/mWWfOIxwPCPXbp4PjSMbBB1vgkCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-12T17:07:33.764549Z","bundle_sha256":"60d3f962ac651690c4714e662c2e1f1bbe6ec0b4883b0ce19faf71c760a6cfd6"}}