{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:JCVXCDDDU2FEJVL6X2N4FV7VDQ","short_pith_number":"pith:JCVXCDDD","schema_version":"1.0","canonical_sha256":"48ab710c63a68a44d57ebe9bc2d7f51c3b1ee21e9779adb3869150381069ee97","source":{"kind":"arxiv","id":"2606.01244","version":1},"attestation_state":"computed","paper":{"title":"Efficient Approximation for Encoder--Decoder Neural Operators via Variation Spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.LG","cs.NA","math.FA","math.NA","math.ST","stat.TH"],"primary_cat":"stat.ML","authors_text":"Jia-Qi Yang, Lei Shi","submitted_at":"2026-05-31T13:53:17Z","abstract_excerpt":"We study operator learning using encoder--decoder neural networks. Inspired by the function-space theory of neural networks, we introduce a variation space as an infinite-dimensional structural class for nonlinear operators. This space is defined through vector-valued measures directly on the input and output spaces. For operators in this space, we establish approximation bounds for encoder--decoder two-layer networks in the Bochner $L^q$ norm. The resulting error bound decomposes into the input encoding error, the output encoding error, and a finite-width approximation term of order $N^{-1/2}"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2606.01244","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"stat.ML","submitted_at":"2026-05-31T13:53:17Z","cross_cats_sorted":["cs.LG","cs.NA","math.FA","math.NA","math.ST","stat.TH"],"title_canon_sha256":"ec6bc3d7935c1bad6d82a2146b7f177270830871edb4c586c3aa7bce7ada6ecc","abstract_canon_sha256":"badb7a79c355b86b952939151616e9e1e0f6a2c340e90019e8964753f9062545"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-02T02:04:27.773863Z","signature_b64":"wNiPDS3rKBD1bnswy0gppyHrMolkugMcf2AO+IBJiIJimfYlYXJUuvMywv7Jih0WNf5ytl2p/7S1vuNBRfPBAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"48ab710c63a68a44d57ebe9bc2d7f51c3b1ee21e9779adb3869150381069ee97","last_reissued_at":"2026-06-02T02:04:27.773428Z","signature_status":"signed_v1","first_computed_at":"2026-06-02T02:04:27.773428Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Efficient Approximation for Encoder--Decoder Neural Operators via Variation Spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.LG","cs.NA","math.FA","math.NA","math.ST","stat.TH"],"primary_cat":"stat.ML","authors_text":"Jia-Qi Yang, Lei Shi","submitted_at":"2026-05-31T13:53:17Z","abstract_excerpt":"We study operator learning using encoder--decoder neural networks. Inspired by the function-space theory of neural networks, we introduce a variation space as an infinite-dimensional structural class for nonlinear operators. This space is defined through vector-valued measures directly on the input and output spaces. For operators in this space, we establish approximation bounds for encoder--decoder two-layer networks in the Bochner $L^q$ norm. The resulting error bound decomposes into the input encoding error, the output encoding error, and a finite-width approximation term of order $N^{-1/2}"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.01244","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/2606.01244/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2606.01244","created_at":"2026-06-02T02:04:27.773488+00:00"},{"alias_kind":"arxiv_version","alias_value":"2606.01244v1","created_at":"2026-06-02T02:04:27.773488+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.01244","created_at":"2026-06-02T02:04:27.773488+00:00"},{"alias_kind":"pith_short_12","alias_value":"JCVXCDDDU2FE","created_at":"2026-06-02T02:04:27.773488+00:00"},{"alias_kind":"pith_short_16","alias_value":"JCVXCDDDU2FEJVL6","created_at":"2026-06-02T02:04:27.773488+00:00"},{"alias_kind":"pith_short_8","alias_value":"JCVXCDDD","created_at":"2026-06-02T02:04:27.773488+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/JCVXCDDDU2FEJVL6X2N4FV7VDQ","json":"https://pith.science/pith/JCVXCDDDU2FEJVL6X2N4FV7VDQ.json","graph_json":"https://pith.science/api/pith-number/JCVXCDDDU2FEJVL6X2N4FV7VDQ/graph.json","events_json":"https://pith.science/api/pith-number/JCVXCDDDU2FEJVL6X2N4FV7VDQ/events.json","paper":"https://pith.science/paper/JCVXCDDD"},"agent_actions":{"view_html":"https://pith.science/pith/JCVXCDDDU2FEJVL6X2N4FV7VDQ","download_json":"https://pith.science/pith/JCVXCDDDU2FEJVL6X2N4FV7VDQ.json","view_paper":"https://pith.science/paper/JCVXCDDD","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2606.01244&json=true","fetch_graph":"https://pith.science/api/pith-number/JCVXCDDDU2FEJVL6X2N4FV7VDQ/graph.json","fetch_events":"https://pith.science/api/pith-number/JCVXCDDDU2FEJVL6X2N4FV7VDQ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/JCVXCDDDU2FEJVL6X2N4FV7VDQ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/JCVXCDDDU2FEJVL6X2N4FV7VDQ/action/storage_attestation","attest_author":"https://pith.science/pith/JCVXCDDDU2FEJVL6X2N4FV7VDQ/action/author_attestation","sign_citation":"https://pith.science/pith/JCVXCDDDU2FEJVL6X2N4FV7VDQ/action/citation_signature","submit_replication":"https://pith.science/pith/JCVXCDDDU2FEJVL6X2N4FV7VDQ/action/replication_record"}},"created_at":"2026-06-02T02:04:27.773488+00:00","updated_at":"2026-06-02T02:04:27.773488+00:00"}