{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:VAWUZREBCSZC2AJ4ZEWGWG5C5Z","short_pith_number":"pith:VAWUZREB","canonical_record":{"source":{"id":"2506.01656","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2025-06-02T13:26:44Z","cross_cats_sorted":["stat.ML"],"title_canon_sha256":"bcc6ee076008a2a301291af8a6d714d7c8f7da021b07f8252b7f6e70aa317891","abstract_canon_sha256":"96d4adf571207ff089f62bda03acaa154ed39318420ca709d6f49c270f453dba"},"schema_version":"1.0"},"canonical_sha256":"a82d4cc48114b22d013cc92c6b1ba2ee509256b0bf25de15dc95997186727e70","source":{"kind":"arxiv","id":"2506.01656","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2506.01656","created_at":"2026-07-05T11:54:47Z"},{"alias_kind":"arxiv_version","alias_value":"2506.01656v2","created_at":"2026-07-05T11:54:47Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2506.01656","created_at":"2026-07-05T11:54:47Z"},{"alias_kind":"pith_short_12","alias_value":"VAWUZREBCSZC","created_at":"2026-07-05T11:54:47Z"},{"alias_kind":"pith_short_16","alias_value":"VAWUZREBCSZC2AJ4","created_at":"2026-07-05T11:54:47Z"},{"alias_kind":"pith_short_8","alias_value":"VAWUZREB","created_at":"2026-07-05T11:54:47Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:VAWUZREBCSZC2AJ4ZEWGWG5C5Z","target":"record","payload":{"canonical_record":{"source":{"id":"2506.01656","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2025-06-02T13:26:44Z","cross_cats_sorted":["stat.ML"],"title_canon_sha256":"bcc6ee076008a2a301291af8a6d714d7c8f7da021b07f8252b7f6e70aa317891","abstract_canon_sha256":"96d4adf571207ff089f62bda03acaa154ed39318420ca709d6f49c270f453dba"},"schema_version":"1.0"},"canonical_sha256":"a82d4cc48114b22d013cc92c6b1ba2ee509256b0bf25de15dc95997186727e70","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:54:47.790146Z","signature_b64":"1smiN1xqQg45xmOLaWf6mTpgZNsu0L+XDBXC1cgcfAAW+cDJsxp4iKY1hwQQvoGxuBcfQzhdLuMEz05XbQV+AQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a82d4cc48114b22d013cc92c6b1ba2ee509256b0bf25de15dc95997186727e70","last_reissued_at":"2026-07-05T11:54:47.789647Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:54:47.789647Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2506.01656","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":"2026-07-05T11:54:47Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"RJNLf5nMtUjKQ+Lm97C4h06+aMdnym3Wk03JN9lRMnjy8J9t7KCWIJ04my87JMOFhUtdQe4/3rB/+U2W4GJCCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-09T16:09:56.744123Z"},"content_sha256":"72ab8bf693a23a3be219479b241ac7b665208befd3dcc02754553a687176eb5e","schema_version":"1.0","event_id":"sha256:72ab8bf693a23a3be219479b241ac7b665208befd3dcc02754553a687176eb5e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:VAWUZREBCSZC2AJ4ZEWGWG5C5Z","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Mixture of Experts Provably Detect and Learn the Latent Cluster Structure in Gradient-Based Learning","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["stat.ML"],"primary_cat":"cs.LG","authors_text":"Kohsei Matsutani, Naoki Nishikawa, Ryotaro Kawata, Taiji Suzuki, Yuri Kinoshita","submitted_at":"2025-06-02T13:26:44Z","abstract_excerpt":"Mixture of Experts (MoE), an ensemble of specialized models equipped with a router that dynamically distributes each input to appropriate experts, has achieved successful results in the field of machine learning. However, theoretical understanding of this architecture is falling behind due to its inherent complexity. In this paper, we theoretically study the sample and runtime complexity of MoE following the stochastic gradient descent (SGD) when learning a regression task with an underlying cluster structure of single index models. On the one hand, we prove that a vanilla neural network fails"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.01656","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/2506.01656/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":"2026-07-05T11:54:47Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+o0jHz06CaoCFOSH7ErxwIVwFuW9xsLiz1urmISsft/b3loJukDol8e+gyIjYouv1ywsUbNpthZaLkIYwv8aBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-09T16:09:56.744623Z"},"content_sha256":"d0d3150d789fac5818c5b93718b4cee7a9bf065889c5c33c7f5b1acee5e2d141","schema_version":"1.0","event_id":"sha256:d0d3150d789fac5818c5b93718b4cee7a9bf065889c5c33c7f5b1acee5e2d141"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/VAWUZREBCSZC2AJ4ZEWGWG5C5Z/bundle.json","state_url":"https://pith.science/pith/VAWUZREBCSZC2AJ4ZEWGWG5C5Z/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/VAWUZREBCSZC2AJ4ZEWGWG5C5Z/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":"2026-08-09T16:09:56Z","links":{"resolver":"https://pith.science/pith/VAWUZREBCSZC2AJ4ZEWGWG5C5Z","bundle":"https://pith.science/pith/VAWUZREBCSZC2AJ4ZEWGWG5C5Z/bundle.json","state":"https://pith.science/pith/VAWUZREBCSZC2AJ4ZEWGWG5C5Z/state.json","well_known_bundle":"https://pith.science/.well-known/pith/VAWUZREBCSZC2AJ4ZEWGWG5C5Z/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:VAWUZREBCSZC2AJ4ZEWGWG5C5Z","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":"96d4adf571207ff089f62bda03acaa154ed39318420ca709d6f49c270f453dba","cross_cats_sorted":["stat.ML"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2025-06-02T13:26:44Z","title_canon_sha256":"bcc6ee076008a2a301291af8a6d714d7c8f7da021b07f8252b7f6e70aa317891"},"schema_version":"1.0","source":{"id":"2506.01656","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2506.01656","created_at":"2026-07-05T11:54:47Z"},{"alias_kind":"arxiv_version","alias_value":"2506.01656v2","created_at":"2026-07-05T11:54:47Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2506.01656","created_at":"2026-07-05T11:54:47Z"},{"alias_kind":"pith_short_12","alias_value":"VAWUZREBCSZC","created_at":"2026-07-05T11:54:47Z"},{"alias_kind":"pith_short_16","alias_value":"VAWUZREBCSZC2AJ4","created_at":"2026-07-05T11:54:47Z"},{"alias_kind":"pith_short_8","alias_value":"VAWUZREB","created_at":"2026-07-05T11:54:47Z"}],"graph_snapshots":[{"event_id":"sha256:d0d3150d789fac5818c5b93718b4cee7a9bf065889c5c33c7f5b1acee5e2d141","target":"graph","created_at":"2026-07-05T11:54:47Z","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/2506.01656/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Mixture of Experts (MoE), an ensemble of specialized models equipped with a router that dynamically distributes each input to appropriate experts, has achieved successful results in the field of machine learning. However, theoretical understanding of this architecture is falling behind due to its inherent complexity. In this paper, we theoretically study the sample and runtime complexity of MoE following the stochastic gradient descent (SGD) when learning a regression task with an underlying cluster structure of single index models. On the one hand, we prove that a vanilla neural network fails","authors_text":"Kohsei Matsutani, Naoki Nishikawa, Ryotaro Kawata, Taiji Suzuki, Yuri Kinoshita","cross_cats":["stat.ML"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2025-06-02T13:26:44Z","title":"Mixture of Experts Provably Detect and Learn the Latent Cluster Structure in Gradient-Based Learning"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.01656","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":"sha256:72ab8bf693a23a3be219479b241ac7b665208befd3dcc02754553a687176eb5e","target":"record","created_at":"2026-07-05T11:54:47Z","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":"96d4adf571207ff089f62bda03acaa154ed39318420ca709d6f49c270f453dba","cross_cats_sorted":["stat.ML"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2025-06-02T13:26:44Z","title_canon_sha256":"bcc6ee076008a2a301291af8a6d714d7c8f7da021b07f8252b7f6e70aa317891"},"schema_version":"1.0","source":{"id":"2506.01656","kind":"arxiv","version":2}},"canonical_sha256":"a82d4cc48114b22d013cc92c6b1ba2ee509256b0bf25de15dc95997186727e70","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a82d4cc48114b22d013cc92c6b1ba2ee509256b0bf25de15dc95997186727e70","first_computed_at":"2026-07-05T11:54:47.789647Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T11:54:47.789647Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"1smiN1xqQg45xmOLaWf6mTpgZNsu0L+XDBXC1cgcfAAW+cDJsxp4iKY1hwQQvoGxuBcfQzhdLuMEz05XbQV+AQ==","signature_status":"signed_v1","signed_at":"2026-07-05T11:54:47.790146Z","signed_message":"canonical_sha256_bytes"},"source_id":"2506.01656","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:72ab8bf693a23a3be219479b241ac7b665208befd3dcc02754553a687176eb5e","sha256:d0d3150d789fac5818c5b93718b4cee7a9bf065889c5c33c7f5b1acee5e2d141"],"state_sha256":"8363ff415bfef582b84547a75f9ccf7e88c4cdceb513e5b4a88301f2aea62c7a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"WoS2cPVsrVfLO4LxZhoVyiv2u+TJqbHGvtZ0a6rR9c/N2tizftYLEHtAPTiVvfrVl4BO/LMzLKeTTUIKCWGLCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-09T16:09:56.748449Z","bundle_sha256":"f62a3c07e349d38f8715a4f590060f1c658d7659cf63b195ed0a738d80a027ad"}}