{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:IPNHHCRXYIGMWP2ILVBWZVBXBL","short_pith_number":"pith:IPNHHCRX","canonical_record":{"source":{"id":"2402.03460","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"stat.ML","submitted_at":"2024-02-05T19:11:57Z","cross_cats_sorted":["cs.LG","cs.NA","cs.NE","math.CO","math.NA"],"title_canon_sha256":"954009c51cbc77979e0ea57c1053372fb465d101c1340c5323d05a40dac6628a","abstract_canon_sha256":"057e98eceeae3d94cae33f689622288b97a66f69b463c8e6bc9547265efb5d1e"},"schema_version":"1.0"},"canonical_sha256":"43da738a37c20ccb3f485d436cd4370ad4ce300faf0f79148c46fc008a5770e4","source":{"kind":"arxiv","id":"2402.03460","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2402.03460","created_at":"2026-07-05T08:23:17Z"},{"alias_kind":"arxiv_version","alias_value":"2402.03460v2","created_at":"2026-07-05T08:23:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2402.03460","created_at":"2026-07-05T08:23:17Z"},{"alias_kind":"pith_short_12","alias_value":"IPNHHCRXYIGM","created_at":"2026-07-05T08:23:17Z"},{"alias_kind":"pith_short_16","alias_value":"IPNHHCRXYIGMWP2I","created_at":"2026-07-05T08:23:17Z"},{"alias_kind":"pith_short_8","alias_value":"IPNHHCRX","created_at":"2026-07-05T08:23:17Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:IPNHHCRXYIGMWP2ILVBWZVBXBL","target":"record","payload":{"canonical_record":{"source":{"id":"2402.03460","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"stat.ML","submitted_at":"2024-02-05T19:11:57Z","cross_cats_sorted":["cs.LG","cs.NA","cs.NE","math.CO","math.NA"],"title_canon_sha256":"954009c51cbc77979e0ea57c1053372fb465d101c1340c5323d05a40dac6628a","abstract_canon_sha256":"057e98eceeae3d94cae33f689622288b97a66f69b463c8e6bc9547265efb5d1e"},"schema_version":"1.0"},"canonical_sha256":"43da738a37c20ccb3f485d436cd4370ad4ce300faf0f79148c46fc008a5770e4","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T08:23:17.042004Z","signature_b64":"wzUBWNCrdA1GCivL0/wxTcVU035wQBwNaKDH0kGuNznqnQRHdNOLNxVj2zVv+5qoGA5ymzpt9I0lW4a1MdNfCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"43da738a37c20ccb3f485d436cd4370ad4ce300faf0f79148c46fc008a5770e4","last_reissued_at":"2026-07-05T08:23:17.041584Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T08:23:17.041584Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2402.03460","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":"2026-07-05T08:23:17Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"BGVTSpKXmjIaaiJ9RQyhCojNMQb8Q3yl7WbP0hkNyylrvlwbuzGWYIr8nKdDl1w7NtLHzYhcrNZNORkmPTB7AQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-06T18:32:09.422351Z"},"content_sha256":"91e76b98581a026f009417e405209ae91e1b6beb6677dcb53364b6e2b3f3c23d","schema_version":"1.0","event_id":"sha256:91e76b98581a026f009417e405209ae91e1b6beb6677dcb53364b6e2b3f3c23d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:IPNHHCRXYIGMWP2ILVBWZVBXBL","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Approximation Rates and VC-Dimension Bounds for (P)ReLU MLP Mixture of Experts","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.LG","cs.NA","cs.NE","math.CO","math.NA"],"primary_cat":"stat.ML","authors_text":"Anastasis Kratsios, Haitz S\\'aez de Oc\\'ariz Borde, Marc T. Law, Takashi Furuya","submitted_at":"2024-02-05T19:11:57Z","abstract_excerpt":"Mixture-of-Experts (MoEs) can scale up beyond traditional deep learning models by employing a routing strategy in which each input is processed by a single \"expert\" deep learning model. This strategy allows us to scale up the number of parameters defining the MoE while maintaining sparse activation, i.e., MoEs only load a small number of their total parameters into GPU VRAM for the forward pass depending on the input. In this paper, we provide an approximation and learning-theoretic analysis of mixtures of expert MLPs with (P)ReLU activation functions. We first prove that for every error level"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2402.03460","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/2402.03460/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":"2026-07-05T08:23:17Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"C6qNVHUapakiOsKZ+06/Ny9NezVBh9jFi1yXdbr9RHleTF5IQWpX2Z0cqkbokB2dzF2u7W6COMtvD+QbAO8eDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-06T18:32:09.422743Z"},"content_sha256":"821c70d77887caf7ea1a4a3eac8d05f8a73b31a01641c5b10de8d1912ebda03a","schema_version":"1.0","event_id":"sha256:821c70d77887caf7ea1a4a3eac8d05f8a73b31a01641c5b10de8d1912ebda03a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/IPNHHCRXYIGMWP2ILVBWZVBXBL/bundle.json","state_url":"https://pith.science/pith/IPNHHCRXYIGMWP2ILVBWZVBXBL/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/IPNHHCRXYIGMWP2ILVBWZVBXBL/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":"2026-07-06T18:32:09Z","links":{"resolver":"https://pith.science/pith/IPNHHCRXYIGMWP2ILVBWZVBXBL","bundle":"https://pith.science/pith/IPNHHCRXYIGMWP2ILVBWZVBXBL/bundle.json","state":"https://pith.science/pith/IPNHHCRXYIGMWP2ILVBWZVBXBL/state.json","well_known_bundle":"https://pith.science/.well-known/pith/IPNHHCRXYIGMWP2ILVBWZVBXBL/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:IPNHHCRXYIGMWP2ILVBWZVBXBL","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":"057e98eceeae3d94cae33f689622288b97a66f69b463c8e6bc9547265efb5d1e","cross_cats_sorted":["cs.LG","cs.NA","cs.NE","math.CO","math.NA"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"stat.ML","submitted_at":"2024-02-05T19:11:57Z","title_canon_sha256":"954009c51cbc77979e0ea57c1053372fb465d101c1340c5323d05a40dac6628a"},"schema_version":"1.0","source":{"id":"2402.03460","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2402.03460","created_at":"2026-07-05T08:23:17Z"},{"alias_kind":"arxiv_version","alias_value":"2402.03460v2","created_at":"2026-07-05T08:23:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2402.03460","created_at":"2026-07-05T08:23:17Z"},{"alias_kind":"pith_short_12","alias_value":"IPNHHCRXYIGM","created_at":"2026-07-05T08:23:17Z"},{"alias_kind":"pith_short_16","alias_value":"IPNHHCRXYIGMWP2I","created_at":"2026-07-05T08:23:17Z"},{"alias_kind":"pith_short_8","alias_value":"IPNHHCRX","created_at":"2026-07-05T08:23:17Z"}],"graph_snapshots":[{"event_id":"sha256:821c70d77887caf7ea1a4a3eac8d05f8a73b31a01641c5b10de8d1912ebda03a","target":"graph","created_at":"2026-07-05T08:23:17Z","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/2402.03460/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Mixture-of-Experts (MoEs) can scale up beyond traditional deep learning models by employing a routing strategy in which each input is processed by a single \"expert\" deep learning model. This strategy allows us to scale up the number of parameters defining the MoE while maintaining sparse activation, i.e., MoEs only load a small number of their total parameters into GPU VRAM for the forward pass depending on the input. In this paper, we provide an approximation and learning-theoretic analysis of mixtures of expert MLPs with (P)ReLU activation functions. We first prove that for every error level","authors_text":"Anastasis Kratsios, Haitz S\\'aez de Oc\\'ariz Borde, Marc T. Law, Takashi Furuya","cross_cats":["cs.LG","cs.NA","cs.NE","math.CO","math.NA"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"stat.ML","submitted_at":"2024-02-05T19:11:57Z","title":"Approximation Rates and VC-Dimension Bounds for (P)ReLU MLP Mixture of Experts"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2402.03460","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":"sha256:91e76b98581a026f009417e405209ae91e1b6beb6677dcb53364b6e2b3f3c23d","target":"record","created_at":"2026-07-05T08:23:17Z","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":"057e98eceeae3d94cae33f689622288b97a66f69b463c8e6bc9547265efb5d1e","cross_cats_sorted":["cs.LG","cs.NA","cs.NE","math.CO","math.NA"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"stat.ML","submitted_at":"2024-02-05T19:11:57Z","title_canon_sha256":"954009c51cbc77979e0ea57c1053372fb465d101c1340c5323d05a40dac6628a"},"schema_version":"1.0","source":{"id":"2402.03460","kind":"arxiv","version":2}},"canonical_sha256":"43da738a37c20ccb3f485d436cd4370ad4ce300faf0f79148c46fc008a5770e4","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"43da738a37c20ccb3f485d436cd4370ad4ce300faf0f79148c46fc008a5770e4","first_computed_at":"2026-07-05T08:23:17.041584Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T08:23:17.041584Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"wzUBWNCrdA1GCivL0/wxTcVU035wQBwNaKDH0kGuNznqnQRHdNOLNxVj2zVv+5qoGA5ymzpt9I0lW4a1MdNfCw==","signature_status":"signed_v1","signed_at":"2026-07-05T08:23:17.042004Z","signed_message":"canonical_sha256_bytes"},"source_id":"2402.03460","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:91e76b98581a026f009417e405209ae91e1b6beb6677dcb53364b6e2b3f3c23d","sha256:821c70d77887caf7ea1a4a3eac8d05f8a73b31a01641c5b10de8d1912ebda03a"],"state_sha256":"560050503a997b2b089af3018fec9df58ff80b101c8c3f7ca7c92eeb0460013a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xRoqYn5WKYExkyIA8n1MAd6d6kIJGhvOprvY8z01VnZP7DxEOtqZ1zerz36lxOCZpU399ivmxPVEURpdtU/ZCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-06T18:32:09.424745Z","bundle_sha256":"c5b448586fef37cbe61d4c58c5d0ad44f45917e8fc28ebcd07b89cd0925309bb"}}