{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/chebyshev-polynomials-moment-matching-and","title":"Chebyshev polynomials, moment matching, and optimal estimation of the unseen","arxiv_id":"1504.01227","date":"2015-04-06","proceeding":null,"authors":["Yihong Wu","Pengkun Yang"],"abstract":"We consider the problem of estimating the support size of a discrete distribution whose minimum non-zero mass is at least $ \\frac{1}{k}$. Under the independent sampling model, we show that the sample complexity, i.e., the minimal sample size to achieve an additive error of $\\epsilon k$ with probability at least 0.1 is within universal constant factors of $ \\frac{k}{\\log k}\\log^2\\frac{1}{\\epsilon} $, which improves the state-of-the-art result of $ \\frac{k}{\\epsilon^2 \\log k} $ in \\cite{VV13}. Similar characterization of the minimax risk is also obtained. Our procedure is a linear estimator based on the Chebyshev polynomial and its approximation-theoretic properties, which can be evaluated in $O(n+\\log^2 k)$ time and attains the sample complexity within a factor of six asymptotically. The superiority of the proposed estimator in terms of accuracy, computational efficiency and scalability is demonstrated in a variety of synthetic and real datasets.","url_abs":"http://arxiv.org/abs/1504.01227v2","url_pdf":"http://arxiv.org/pdf/1504.01227v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"chebyshev-polynomials-moment-matching-and","repo_url":"https://github.com/Albuso0/support","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":"https://syntology.ai/paper/1504.01227","atlas_url":"https://app.syntology.ai/?focus=1504.01227","mcp":{"get_harvested_code_for_paper":{"arxiv_id":"1504.01227"}},"developers":"https://syntology.ai/developers","read_at":"2026-09-25T09:33:49+00:00","read_at_is":"when the build read Syntology's graph, not when any sample ran","claim":"Per-sample execution status on synthesized fixtures; not a correctness claim about the paper. 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