{"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/tensor-robust-principal-component-analysis","title":"Tensor Robust Principal Component Analysis with A New Tensor Nuclear Norm","arxiv_id":"1804.03728","date":"2018-04-10","proceeding":null,"authors":["Canyi Lu","Jiashi Feng","Yudong Chen","Wei Liu","Zhouchen Lin","Shuicheng Yan"],"abstract":"In this paper, we consider the Tensor Robust Principal Component Analysis\n(TRPCA) problem, which aims to exactly recover the low-rank and sparse\ncomponents from their sum. Our model is based on the recently proposed\ntensor-tensor product (or t-product). Induced by the t-product, we first\nrigorously deduce the tensor spectral norm, tensor nuclear norm, and tensor\naverage rank, and show that the tensor nuclear norm is the convex envelope of\nthe tensor average rank within the unit ball of the tensor spectral norm. These\ndefinitions, their relationships and properties are consistent with matrix\ncases. Equipped with the new tensor nuclear norm, we then solve the TRPCA\nproblem by solving a convex program and provide the theoretical guarantee for\nthe exact recovery. Our TRPCA model and recovery guarantee include matrix RPCA\nas a special case. Numerical experiments verify our results, and the\napplications to image recovery and background modeling problems demonstrate the\neffectiveness of our method.","url_abs":"http://arxiv.org/abs/1804.03728v2","url_pdf":"http://arxiv.org/pdf/1804.03728v2.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":"tensor-robust-principal-component-analysis","repo_url":"https://github.com/zhaoxile/reproducible-tensor-completion-state-of-the-art","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":{"status":"ok"}}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":"https://syntology.ai/paper/1804.03728","atlas_url":"https://app.syntology.ai/?focus=1804.03728","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}