{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2021:BOPK7LMOCFGD2NCEW6SNWSNWJC","short_pith_number":"pith:BOPK7LMO","canonical_record":{"source":{"id":"2102.11742","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2021-02-23T15:10:15Z","cross_cats_sorted":["cond-mat.dis-nn","cond-mat.stat-mech","stat.ML"],"title_canon_sha256":"428267565f764caa08049a677248e85435a37ab4c097d5cb522d84ea4ba6b138","abstract_canon_sha256":"75b569a049ff640658cb2260ca8fbb82d0205a823aa44627d88ab6e0be0b5454"},"schema_version":"1.0"},"canonical_sha256":"0b9eafad8e114c3d3444b7a4db49b648a2ea726dda68a40f2fbb5097a9f07192","source":{"kind":"arxiv","id":"2102.11742","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2102.11742","created_at":"2026-07-05T02:48:10Z"},{"alias_kind":"arxiv_version","alias_value":"2102.11742v2","created_at":"2026-07-05T02:48:10Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2102.11742","created_at":"2026-07-05T02:48:10Z"},{"alias_kind":"pith_short_12","alias_value":"BOPK7LMOCFGD","created_at":"2026-07-05T02:48:10Z"},{"alias_kind":"pith_short_16","alias_value":"BOPK7LMOCFGD2NCE","created_at":"2026-07-05T02:48:10Z"},{"alias_kind":"pith_short_8","alias_value":"BOPK7LMO","created_at":"2026-07-05T02:48:10Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2021:BOPK7LMOCFGD2NCEW6SNWSNWJC","target":"record","payload":{"canonical_record":{"source":{"id":"2102.11742","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2021-02-23T15:10:15Z","cross_cats_sorted":["cond-mat.dis-nn","cond-mat.stat-mech","stat.ML"],"title_canon_sha256":"428267565f764caa08049a677248e85435a37ab4c097d5cb522d84ea4ba6b138","abstract_canon_sha256":"75b569a049ff640658cb2260ca8fbb82d0205a823aa44627d88ab6e0be0b5454"},"schema_version":"1.0"},"canonical_sha256":"0b9eafad8e114c3d3444b7a4db49b648a2ea726dda68a40f2fbb5097a9f07192","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T02:48:10.493142Z","signature_b64":"0z6XfMv4LSRJpw/2LDNX5Xw64nT0/v0SgDhh2zmkESXSRpeqs2NAi/3CWfpoqdncGbyKIYxszdhNO7lXIcY5BQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0b9eafad8e114c3d3444b7a4db49b648a2ea726dda68a40f2fbb5097a9f07192","last_reissued_at":"2026-07-05T02:48:10.492694Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T02:48:10.492694Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2102.11742","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":"2026-07-05T02:48:10Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"V7dfoiTXELlOd3uft7uh0jpZ4ddIzUwsaEo2jxmM4qwooiL/jJMoDIQI2UbvBvA/7Z88TbzmkgYhJYww1znLDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-09T03:56:36.061819Z"},"content_sha256":"6defddd93aa7e85d7880baa21c77bb5af0b33133d03f8fba6dbd852ad7f84f6e","schema_version":"1.0","event_id":"sha256:6defddd93aa7e85d7880baa21c77bb5af0b33133d03f8fba6dbd852ad7f84f6e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2021:BOPK7LMOCFGD2NCEW6SNWSNWJC","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Classifying high-dimensional Gaussian mixtures: Where kernel methods fail and neural networks succeed","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cond-mat.dis-nn","cond-mat.stat-mech","stat.ML"],"primary_cat":"cs.LG","authors_text":"Florent Krzakala, Lenka Zdeborov\\'a, Maria Refinetti, Sebastian Goldt","submitted_at":"2021-02-23T15:10:15Z","abstract_excerpt":"A recent series of theoretical works showed that the dynamics of neural networks with a certain initialisation are well-captured by kernel methods. Concurrent empirical work demonstrated that kernel methods can come close to the performance of neural networks on some image classification tasks. These results raise the question of whether neural networks only learn successfully if kernels also learn successfully, despite neural networks being more expressive. Here, we show theoretically that two-layer neural networks (2LNN) with only a few hidden neurons can beat the performance of kernel learn"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2102.11742","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/2102.11742/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":"2026-07-05T02:48:10Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mUhNzSEwGNfS4hKq/sboIj/KWwF68XEtvgA0Lx3hOhI6AUDwZJO9JFvLnkieahPsMBOi/qskiSvce5A9NV/KDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-09T03:56:36.062194Z"},"content_sha256":"4d6340f0ff7d51b9ed0dc87e2e14bb8bf5f12f7b1c60b78f53afc0c4ede29da4","schema_version":"1.0","event_id":"sha256:4d6340f0ff7d51b9ed0dc87e2e14bb8bf5f12f7b1c60b78f53afc0c4ede29da4"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/BOPK7LMOCFGD2NCEW6SNWSNWJC/bundle.json","state_url":"https://pith.science/pith/BOPK7LMOCFGD2NCEW6SNWSNWJC/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/BOPK7LMOCFGD2NCEW6SNWSNWJC/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":"2026-07-09T03:56:36Z","links":{"resolver":"https://pith.science/pith/BOPK7LMOCFGD2NCEW6SNWSNWJC","bundle":"https://pith.science/pith/BOPK7LMOCFGD2NCEW6SNWSNWJC/bundle.json","state":"https://pith.science/pith/BOPK7LMOCFGD2NCEW6SNWSNWJC/state.json","well_known_bundle":"https://pith.science/.well-known/pith/BOPK7LMOCFGD2NCEW6SNWSNWJC/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2021:BOPK7LMOCFGD2NCEW6SNWSNWJC","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":"75b569a049ff640658cb2260ca8fbb82d0205a823aa44627d88ab6e0be0b5454","cross_cats_sorted":["cond-mat.dis-nn","cond-mat.stat-mech","stat.ML"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2021-02-23T15:10:15Z","title_canon_sha256":"428267565f764caa08049a677248e85435a37ab4c097d5cb522d84ea4ba6b138"},"schema_version":"1.0","source":{"id":"2102.11742","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2102.11742","created_at":"2026-07-05T02:48:10Z"},{"alias_kind":"arxiv_version","alias_value":"2102.11742v2","created_at":"2026-07-05T02:48:10Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2102.11742","created_at":"2026-07-05T02:48:10Z"},{"alias_kind":"pith_short_12","alias_value":"BOPK7LMOCFGD","created_at":"2026-07-05T02:48:10Z"},{"alias_kind":"pith_short_16","alias_value":"BOPK7LMOCFGD2NCE","created_at":"2026-07-05T02:48:10Z"},{"alias_kind":"pith_short_8","alias_value":"BOPK7LMO","created_at":"2026-07-05T02:48:10Z"}],"graph_snapshots":[{"event_id":"sha256:4d6340f0ff7d51b9ed0dc87e2e14bb8bf5f12f7b1c60b78f53afc0c4ede29da4","target":"graph","created_at":"2026-07-05T02:48:10Z","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/2102.11742/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"A recent series of theoretical works showed that the dynamics of neural networks with a certain initialisation are well-captured by kernel methods. Concurrent empirical work demonstrated that kernel methods can come close to the performance of neural networks on some image classification tasks. These results raise the question of whether neural networks only learn successfully if kernels also learn successfully, despite neural networks being more expressive. Here, we show theoretically that two-layer neural networks (2LNN) with only a few hidden neurons can beat the performance of kernel learn","authors_text":"Florent Krzakala, Lenka Zdeborov\\'a, Maria Refinetti, Sebastian Goldt","cross_cats":["cond-mat.dis-nn","cond-mat.stat-mech","stat.ML"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2021-02-23T15:10:15Z","title":"Classifying high-dimensional Gaussian mixtures: Where kernel methods fail and neural networks succeed"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2102.11742","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":"sha256:6defddd93aa7e85d7880baa21c77bb5af0b33133d03f8fba6dbd852ad7f84f6e","target":"record","created_at":"2026-07-05T02:48:10Z","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":"75b569a049ff640658cb2260ca8fbb82d0205a823aa44627d88ab6e0be0b5454","cross_cats_sorted":["cond-mat.dis-nn","cond-mat.stat-mech","stat.ML"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2021-02-23T15:10:15Z","title_canon_sha256":"428267565f764caa08049a677248e85435a37ab4c097d5cb522d84ea4ba6b138"},"schema_version":"1.0","source":{"id":"2102.11742","kind":"arxiv","version":2}},"canonical_sha256":"0b9eafad8e114c3d3444b7a4db49b648a2ea726dda68a40f2fbb5097a9f07192","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0b9eafad8e114c3d3444b7a4db49b648a2ea726dda68a40f2fbb5097a9f07192","first_computed_at":"2026-07-05T02:48:10.492694Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T02:48:10.492694Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"0z6XfMv4LSRJpw/2LDNX5Xw64nT0/v0SgDhh2zmkESXSRpeqs2NAi/3CWfpoqdncGbyKIYxszdhNO7lXIcY5BQ==","signature_status":"signed_v1","signed_at":"2026-07-05T02:48:10.493142Z","signed_message":"canonical_sha256_bytes"},"source_id":"2102.11742","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6defddd93aa7e85d7880baa21c77bb5af0b33133d03f8fba6dbd852ad7f84f6e","sha256:4d6340f0ff7d51b9ed0dc87e2e14bb8bf5f12f7b1c60b78f53afc0c4ede29da4"],"state_sha256":"86de786aae8a240209eb3e297ff73a7e8b0a8074f7bf398560cd3fed18fcff2d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"dqnSKrGVThXwmzgy+h3//UVYfE3g+D8AA2NyyIfbTLty3M1U5GWSdVIDPicUzirMfNNP7Ou7WVWZUcPVk614DQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-09T03:56:36.064244Z","bundle_sha256":"834e03ce5a379e5494d5a8ee20411b85debcdb0a0e856f89b383ecef9bda59a4"}}