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The index u is a subset of {1,2,…,d+1}, where each term f_{u,ξ} depends only on the variables with indices in u."},{"attestation":"unclaimed","claim_id":"C2","kind":"weakest_assumption","source":"verdict.weakest_assumption","status":"machine_extracted","text":"The natural geometry on the sphere naturally leads to the dependencies between the input variables, and suitable orthogonal basis functions exist that permit the decomposition and approximation."},{"attestation":"unclaimed","claim_id":"C3","kind":"one_line_summary","source":"verdict.one_line_summary","status":"machine_extracted","text":"A parity-augmented ANOVA decomposition is established for functions on the sphere using orthogonal bases to capture geometry-induced variable dependencies."},{"attestation":"unclaimed","claim_id":"C4","kind":"headline","source":"verdict.pith_extraction.headline","status":"machine_extracted","text":"A function on the sphere decomposes into a sum of terms each depending on a coordinate subset and a parity vector."}],"snapshot_sha256":"43bf55e48963c9f5135caa7453382b4ca68d022590202f352dceb96d43294e87"},"formal_canon":{"evidence_count":2,"snapshot_sha256":"60613f69fcc73fb8f25f53e831ecca784b6681502f8b4dfc1e6db33a407ede03"},"paper":{"abstract_excerpt":"We establish sensitivity analysis on the sphere. 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