{"paper":{"title":"Optimal Sampling for Kernel Quadrature on Unbounded Domains","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Constructs an n-dependent kernel-agnostic sampling distribution achieving minimax worst-case error rates for quadrature over smoothness classes on unbounded domains.","cross_cats":["stat.ME"],"primary_cat":"stat.CO","authors_text":"Christian Robert (CEREMADE), Edoardo Bandoni (CEREMADE), Julien Stoehr (CEREMADE)","submitted_at":"2026-05-18T09:39:11Z","abstract_excerpt":"Kernel quadrature is widely used to approximate integrals of smooth functions, with worst-case error typically decaying at the minimax rate $n^{-\\alpha/d}$ for smoothness $\\alpha$ in dimension $d$. Existing rate-optimal methods often depend on deterministic point sets tailored to a specific kernel, making them sensitive to misspecification and less robust in practice. In this work, we study randomized quadrature methods with a focus on robustness rather than kernel-specific optimality. We construct an explicit, $n$-dependent sampling distribution that achieves minimax rates for worst-case erro"},"claims":{"count":3,"items":[{"kind":"strongest_claim","text":"We construct an explicit, n-dependent sampling distribution that achieves minimax rates for worst-case error over smoothness classes without requiring knowledge of the kernel.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The existence and explicit constructibility of an n-dependent sampling distribution whose worst-case error matches the minimax rate over the smoothness class, independent of the kernel, for the chosen unbounded measures (Gaussian, Student-t).","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Constructs an n-dependent kernel-agnostic sampling distribution achieving minimax worst-case error rates for quadrature over smoothness classes on unbounded domains.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"}],"snapshot_sha256":"47019c14211ef73069d8e986bca5cf128a52e59b39006f8396cb7d567894ebf5"},"source":{"id":"2605.18134","kind":"arxiv","version":1},"verdict":{"id":"e749e5d8-3881-4cc6-8e45-05871a3dc8f2","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-20T00:00:43.490572Z","strongest_claim":"We construct an explicit, n-dependent sampling distribution that achieves minimax rates for worst-case error over smoothness classes without requiring knowledge of the kernel.","one_line_summary":"Constructs an n-dependent kernel-agnostic sampling distribution achieving minimax worst-case error rates for quadrature over smoothness classes on unbounded domains.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The existence and explicit constructibility of an n-dependent sampling distribution whose worst-case error matches the minimax rate over the smoothness class, independent of the kernel, for the chosen unbounded measures (Gaussian, Student-t).","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.18134/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"claim_evidence","ran_at":"2026-05-19T23:41:59.121219Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T23:33:35.389712Z","status":"skipped","version":"1.0.0","findings_count":0}],"snapshot_sha256":"7ce35f2bae2697c8150db4a92dedd993422fed10aba3ecb6f4a010284bdaa481"},"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"}