{"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/concentration-study-of-m-estimators-using-the","title":"Concentration study of M-estimators using the influence function","arxiv_id":"2104.04416","date":"2021-04-09","proceeding":null,"authors":["Timothée Mathieu"],"abstract":"We present a new finite-sample analysis of M-estimators of locations in $\\mathbb{R}^d$ using the tool of the influence function. In particular, we show that the deviations of an M-estimator can be controlled thanks to its influence function (or its score function) and then, we use concentration inequality on M-estimators to investigate the robust estimation of the mean in high dimension in a corrupted setting (adversarial corruption setting) for bounded and unbounded score functions. For a sample of size $n$ and covariance matrix $\\Sigma$, we attain the minimax speed $\\sqrt{Tr(\\Sigma)/n}+\\sqrt{\\|\\Sigma\\|_{op}\\log(1/\\delta)/n}$ with probability larger than $1-\\delta$ in a heavy-tailed setting. One of the major advantages of our approach compared to others recently proposed is that our estimator is tractable and fast to compute even in very high dimension with a complexity of $O(nd\\log(Tr(\\Sigma)))$ where $n$ is the sample size and $\\Sigma$ is the covariance matrix of the inliers. In practice, the code that we make available for this article proves to be very fast.","url_abs":"https://arxiv.org/abs/2104.04416v4","url_pdf":"https://arxiv.org/pdf/2104.04416v4.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":"concentration-study-of-m-estimators-using-the","repo_url":"https://github.com/TimotheeMathieu/RobustMeanEstimator","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}