{"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/near-optimal-sample-complexity-for-convex","title":"Near-optimal sample complexity for convex tensor completion","arxiv_id":"1711.04965","date":"2017-11-14","proceeding":null,"authors":["Navid Ghadermarzy","Yaniv Plan","Özgür Yılmaz"],"abstract":"We analyze low rank tensor completion (TC) using noisy measurements of a\nsubset of the tensor. Assuming a rank-$r$, order-$d$, $N \\times N \\times \\cdots\n\\times N$ tensor where $r=O(1)$, the best sampling complexity that was achieved\nis $O(N^{\\frac{d}{2}})$, which is obtained by solving a tensor nuclear-norm\nminimization problem. However, this bound is significantly larger than the\nnumber of free variables in a low rank tensor which is $O(dN)$. In this paper,\nwe show that by using an atomic-norm whose atoms are rank-$1$ sign tensors, one\ncan obtain a sample complexity of $O(dN)$. Moreover, we generalize the matrix\nmax-norm definition to tensors, which results in a max-quasi-norm (max-qnorm)\nwhose unit ball has small Rademacher complexity. We prove that solving a\nconstrained least squares estimation using either the convex atomic-norm or the\nnonconvex max-qnorm results in optimal sample complexity for the problem of\nlow-rank tensor completion. Furthermore, we show that these bounds are nearly\nminimax rate-optimal. We also provide promising numerical results for max-qnorm\nconstrained tensor completion, showing improved recovery results compared to\nmatricization and alternating least squares.","url_abs":"http://arxiv.org/abs/1711.04965v1","url_pdf":"http://arxiv.org/pdf/1711.04965v1.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":"near-optimal-sample-complexity-for-convex","repo_url":"https://github.com/navidghadermarzy/TensorCompletion_1bit_noisy","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":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}