{"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/algebraic-variety-models-for-high-rank-matrix","title":"Algebraic Variety Models for High-Rank Matrix Completion","arxiv_id":"1703.09631","date":"2017-03-28","proceeding":"ICML 2017 8","authors":["Greg Ongie","Rebecca Willett","Robert D. Nowak","Laura Balzano"],"abstract":"We consider a generalization of low-rank matrix completion to the case where\nthe data belongs to an algebraic variety, i.e. each data point is a solution to\na system of polynomial equations. In this case the original matrix is possibly\nhigh-rank, but it becomes low-rank after mapping each column to a higher\ndimensional space of monomial features. Many well-studied extensions of linear\nmodels, including affine subspaces and their union, can be described by a\nvariety model. In addition, varieties can be used to model a richer class of\nnonlinear quadratic and higher degree curves and surfaces. We study the\nsampling requirements for matrix completion under a variety model with a focus\non a union of affine subspaces. We also propose an efficient matrix completion\nalgorithm that minimizes a convex or non-convex surrogate of the rank of the\nmatrix of monomial features. Our algorithm uses the well-known \"kernel trick\"\nto avoid working directly with the high-dimensional monomial matrix. We show\nthe proposed algorithm is able to recover synthetically generated data up to\nthe predicted sampling complexity bounds. The proposed algorithm also\noutperforms standard low rank matrix completion and subspace clustering\ntechniques in experiments with real data.","url_abs":"http://arxiv.org/abs/1703.09631v1","url_pdf":"http://arxiv.org/pdf/1703.09631v1.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":"algebraic-variety-models-for-high-rank-matrix","repo_url":"https://github.com/gregongie/vmc","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":{"status":"ok","spdx":"MIT"}}],"tasks":[{"task_slug":"clustering","task_name":"Clustering"},{"task_slug":"low-rank-matrix-completion","task_name":"Low-Rank Matrix Completion"},{"task_slug":"matrix-completion","task_name":"Matrix Completion"},{"task_slug":"high","task_name":"Vocal Bursts Intensity Prediction"}],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=1703.09631","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}