{"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/hutch-optimal-stochastic-trace-estimation","title":"Hutch++: Optimal Stochastic Trace Estimation","arxiv_id":"2010.09649","date":"2020-10-19","proceeding":null,"authors":["Raphael A. Meyer","Cameron Musco","Christopher Musco","David P. Woodruff"],"abstract":"We study the problem of estimating the trace of a matrix $A$ that can only be accessed through matrix-vector multiplication. We introduce a new randomized algorithm, Hutch++, which computes a $(1 \\pm \\epsilon)$ approximation to $tr(A)$ for any positive semidefinite (PSD) $A$ using just $O(1/\\epsilon)$ matrix-vector products. This improves on the ubiquitous Hutchinson's estimator, which requires $O(1/\\epsilon^2)$ matrix-vector products. Our approach is based on a simple technique for reducing the variance of Hutchinson's estimator using a low-rank approximation step, and is easy to implement and analyze. Moreover, we prove that, up to a logarithmic factor, the complexity of Hutch++ is optimal amongst all matrix-vector query algorithms, even when queries can be chosen adaptively. We show that it significantly outperforms Hutchinson's method in experiments. While our theory mainly requires $A$ to be positive semidefinite, we provide generalized guarantees for general square matrices, and show empirical gains in such applications.","url_abs":"https://arxiv.org/abs/2010.09649v5","url_pdf":"https://arxiv.org/pdf/2010.09649v5.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":"abstracts"},"code_links":[{"paper_slug":"hutch-optimal-stochastic-trace-estimation","repo_url":"https://github.com/RaphaelArkadyMeyerNYU/HutchPlusPlus","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":{"status":"ok","spdx":"MIT"}}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=2010.09649","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}