{"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/matrix-norm-estimation-from-a-few-entries","title":"Matrix Norm Estimation from a Few Entries","arxiv_id":null,"date":"2017-12-01","proceeding":"NeurIPS 2017 12","authors":["Ashish Khetan","Sewoong Oh"],"abstract":"Singular values of a data in a matrix form provide insights on the structure of the data, the effective dimensionality, and the choice of hyper-parameters on higher-level data analysis tools. However, in many practical applications such as collaborative filtering and network analysis,  we only get a partial observation. Under such scenarios, we consider the fundamental problem of recovering various spectral properties of the underlying matrix from a sampling of its entries. We propose a framework of first estimating the Schatten $k$-norms of a matrix for several values of $k$, and using these as surrogates for estimating spectral properties of interest, such as   the spectrum itself or the rank.  This paper focuses on the technical challenges in accurately estimating the Schatten norms from a sampling of a matrix. We introduce  a novel unbiased estimator based on counting small structures in a graph and provide  guarantees that match its empirical performances.  Our theoretical analysis shows that Schatten norms can be recovered accurately from strictly smaller number of samples compared to what is needed to recover the underlying low-rank matrix. Numerical experiments suggest that we significantly improve upon a competing approach of using matrix completion methods.","url_abs":"http://papers.nips.cc/paper/7221-matrix-norm-estimation-from-a-few-entries","url_pdf":"http://papers.nips.cc/paper/7221-matrix-norm-estimation-from-a-few-entries.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":"matrix-norm-estimation-from-a-few-entries","repo_url":"https://github.com/khetan2/Schatten_norm_estimation","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[{"task_slug":"collaborative-filtering","task_name":"Collaborative Filtering"},{"task_slug":"matrix-completion","task_name":"Matrix Completion"}],"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}