{"paper":{"title":"A new upper bound for sampling numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.NA"],"primary_cat":"math.NA","authors_text":"Martin Sch\\\"afer, Nicolas Nagel, Tino Ullrich","submitted_at":"2020-09-30T08:32:18Z","abstract_excerpt":"We provide a new upper bound for sampling numbers $(g_n)_{n\\in \\mathbb{N}}$ associated to the compact embedding of a separable reproducing kernel Hilbert space into the space of square integrable functions. There are universal constants $C,c>0$ (which are specified in the paper) such that $$\n  g^2_n \\leq \\frac{C\\log(n)}{n}\\sum\\limits_{k\\geq \\lfloor cn \\rfloor} \\sigma_k^2\\quad,\\quad n\\geq 2\\,, $$ where $(\\sigma_k)_{k\\in \\mathbb{N}}$ is the sequence of singular numbers (approximation numbers) of the Hilbert-Schmidt embedding $\\text{Id}:H(K) \\to L_2(D,\\varrho_D)$. The algorithm which realizes the"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2010.00327","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/2010.00327/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"}