{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:DV5JSGR2ICEKKBTT56VHDLIKNC","short_pith_number":"pith:DV5JSGR2","canonical_record":{"source":{"id":"2605.22724","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2026-05-21T16:57:33Z","cross_cats_sorted":["cs.NA","math.NA","stat.ML"],"title_canon_sha256":"7767bcc5476d19e0e2c273db73eab76b47c7b0c58ab2e0ef39e137506400b7a3","abstract_canon_sha256":"79eef1d802f2b15f93cc656d5a06a4d350e9a5dd37bda1bdfbc7c94292c2d946"},"schema_version":"1.0"},"canonical_sha256":"1d7a991a3a4088a50673efaa71ad0a689a781c1171ca23a6c4f770c0666af422","source":{"kind":"arxiv","id":"2605.22724","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.22724","created_at":"2026-05-22T02:04:52Z"},{"alias_kind":"arxiv_version","alias_value":"2605.22724v1","created_at":"2026-05-22T02:04:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.22724","created_at":"2026-05-22T02:04:52Z"},{"alias_kind":"pith_short_12","alias_value":"DV5JSGR2ICEK","created_at":"2026-05-22T02:04:52Z"},{"alias_kind":"pith_short_16","alias_value":"DV5JSGR2ICEKKBTT","created_at":"2026-05-22T02:04:52Z"},{"alias_kind":"pith_short_8","alias_value":"DV5JSGR2","created_at":"2026-05-22T02:04:52Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:DV5JSGR2ICEKKBTT56VHDLIKNC","target":"record","payload":{"canonical_record":{"source":{"id":"2605.22724","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2026-05-21T16:57:33Z","cross_cats_sorted":["cs.NA","math.NA","stat.ML"],"title_canon_sha256":"7767bcc5476d19e0e2c273db73eab76b47c7b0c58ab2e0ef39e137506400b7a3","abstract_canon_sha256":"79eef1d802f2b15f93cc656d5a06a4d350e9a5dd37bda1bdfbc7c94292c2d946"},"schema_version":"1.0"},"canonical_sha256":"1d7a991a3a4088a50673efaa71ad0a689a781c1171ca23a6c4f770c0666af422","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-22T02:04:52.044010Z","signature_b64":"av8ddQoTVzLJHKIyqkLP7EokmwxkLu0ft+gr4kh/p18Sy5vcfe830gYW4krdnAWY6AVT1968O/wfp9f/xdZoDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1d7a991a3a4088a50673efaa71ad0a689a781c1171ca23a6c4f770c0666af422","last_reissued_at":"2026-05-22T02:04:52.043411Z","signature_status":"signed_v1","first_computed_at":"2026-05-22T02:04:52.043411Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2605.22724","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":"2026-05-22T02:04:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3v6zse3jlWjMUlVNKWPV+Ek/pLzJWv1+yZb9CDYFWM3IdHkntXi8UHy5xF3s7jv7iyisek+Ey6/en7XxHiESBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T13:30:57.550214Z"},"content_sha256":"4cd0f727b11ce308886fad6d168d780e77958cbcafb269f9b1cee8affaa4be19","schema_version":"1.0","event_id":"sha256:4cd0f727b11ce308886fad6d168d780e77958cbcafb269f9b1cee8affaa4be19"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:DV5JSGR2ICEKKBTT56VHDLIKNC","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Multiple Neural Operators Achieve Near-Optimal Rates for Multi-Task Learning","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.NA","math.NA","stat.ML"],"primary_cat":"cs.LG","authors_text":"Adrien Weihs, Hayden Schaeffer","submitted_at":"2026-05-21T16:57:33Z","abstract_excerpt":"We study the approximation and statistical complexity of learning collections of operators in a shared multi-task setting, with a focus on the Multiple Neural Operators (MNO) architecture. For broad classes of Lipschitz multiple operator maps, we derive near-optimal upper bounds for approximation and statistical generalization. On the lower-bound side, we establish a curse of parametric complexity and prove corresponding minimax rates. Together, these results show that shared representations across tasks do not increase the overall cost: multi-task operator learning follows the same scaling la"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.22724","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/2605.22724/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":"2026-05-22T02:04:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"UUzwGjsGQm6l4HbwYbZnfbcnf2V3InpGpZ25ruToxyk6ZM/f3hSViAmrjynWmcBt3a9oVvqqlguNVsijcHSeAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T13:30:57.550691Z"},"content_sha256":"cd0401120ac99ae41203707df1b36eb8221f49c9b467d5e6f35df0318cb2032b","schema_version":"1.0","event_id":"sha256:cd0401120ac99ae41203707df1b36eb8221f49c9b467d5e6f35df0318cb2032b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/DV5JSGR2ICEKKBTT56VHDLIKNC/bundle.json","state_url":"https://pith.science/pith/DV5JSGR2ICEKKBTT56VHDLIKNC/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/DV5JSGR2ICEKKBTT56VHDLIKNC/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":"2026-05-27T13:30:57Z","links":{"resolver":"https://pith.science/pith/DV5JSGR2ICEKKBTT56VHDLIKNC","bundle":"https://pith.science/pith/DV5JSGR2ICEKKBTT56VHDLIKNC/bundle.json","state":"https://pith.science/pith/DV5JSGR2ICEKKBTT56VHDLIKNC/state.json","well_known_bundle":"https://pith.science/.well-known/pith/DV5JSGR2ICEKKBTT56VHDLIKNC/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:DV5JSGR2ICEKKBTT56VHDLIKNC","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":"79eef1d802f2b15f93cc656d5a06a4d350e9a5dd37bda1bdfbc7c94292c2d946","cross_cats_sorted":["cs.NA","math.NA","stat.ML"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2026-05-21T16:57:33Z","title_canon_sha256":"7767bcc5476d19e0e2c273db73eab76b47c7b0c58ab2e0ef39e137506400b7a3"},"schema_version":"1.0","source":{"id":"2605.22724","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.22724","created_at":"2026-05-22T02:04:52Z"},{"alias_kind":"arxiv_version","alias_value":"2605.22724v1","created_at":"2026-05-22T02:04:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.22724","created_at":"2026-05-22T02:04:52Z"},{"alias_kind":"pith_short_12","alias_value":"DV5JSGR2ICEK","created_at":"2026-05-22T02:04:52Z"},{"alias_kind":"pith_short_16","alias_value":"DV5JSGR2ICEKKBTT","created_at":"2026-05-22T02:04:52Z"},{"alias_kind":"pith_short_8","alias_value":"DV5JSGR2","created_at":"2026-05-22T02:04:52Z"}],"graph_snapshots":[{"event_id":"sha256:cd0401120ac99ae41203707df1b36eb8221f49c9b467d5e6f35df0318cb2032b","target":"graph","created_at":"2026-05-22T02:04:52Z","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/2605.22724/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We study the approximation and statistical complexity of learning collections of operators in a shared multi-task setting, with a focus on the Multiple Neural Operators (MNO) architecture. For broad classes of Lipschitz multiple operator maps, we derive near-optimal upper bounds for approximation and statistical generalization. On the lower-bound side, we establish a curse of parametric complexity and prove corresponding minimax rates. Together, these results show that shared representations across tasks do not increase the overall cost: multi-task operator learning follows the same scaling la","authors_text":"Adrien Weihs, Hayden Schaeffer","cross_cats":["cs.NA","math.NA","stat.ML"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2026-05-21T16:57:33Z","title":"Multiple Neural Operators Achieve Near-Optimal Rates for Multi-Task Learning"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.22724","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":"sha256:4cd0f727b11ce308886fad6d168d780e77958cbcafb269f9b1cee8affaa4be19","target":"record","created_at":"2026-05-22T02:04:52Z","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":"79eef1d802f2b15f93cc656d5a06a4d350e9a5dd37bda1bdfbc7c94292c2d946","cross_cats_sorted":["cs.NA","math.NA","stat.ML"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2026-05-21T16:57:33Z","title_canon_sha256":"7767bcc5476d19e0e2c273db73eab76b47c7b0c58ab2e0ef39e137506400b7a3"},"schema_version":"1.0","source":{"id":"2605.22724","kind":"arxiv","version":1}},"canonical_sha256":"1d7a991a3a4088a50673efaa71ad0a689a781c1171ca23a6c4f770c0666af422","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1d7a991a3a4088a50673efaa71ad0a689a781c1171ca23a6c4f770c0666af422","first_computed_at":"2026-05-22T02:04:52.043411Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-22T02:04:52.043411Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"av8ddQoTVzLJHKIyqkLP7EokmwxkLu0ft+gr4kh/p18Sy5vcfe830gYW4krdnAWY6AVT1968O/wfp9f/xdZoDg==","signature_status":"signed_v1","signed_at":"2026-05-22T02:04:52.044010Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.22724","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4cd0f727b11ce308886fad6d168d780e77958cbcafb269f9b1cee8affaa4be19","sha256:cd0401120ac99ae41203707df1b36eb8221f49c9b467d5e6f35df0318cb2032b"],"state_sha256":"23421d2aa9f3839fef609c46c310fbf2d82c41bc604c1067f7f926760168b01d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FAfRQ9UrSciDfyb9l04h+FpVhhdNd4cVHoPGiHLe7oarR5AnbyNCSNufBl3CAMjTCfgtmlEwdRXIpoOvKu+LDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T13:30:57.554067Z","bundle_sha256":"d7d1f6949b0a8eca57da5559682196c55b3dad342468484491b28f012733bbac"}}