{"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/exact-low-tubal-rank-tensor-recovery-from","title":"Exact Low Tubal Rank Tensor Recovery from Gaussian Measurements","arxiv_id":"1806.02511","date":"2018-06-07","proceeding":null,"authors":["Canyi Lu","Jiashi Feng","Zhouchen Lin","Shuicheng Yan"],"abstract":"The recent proposed Tensor Nuclear Norm (TNN) [Lu et al., 2016; 2018a] is an\ninteresting convex penalty induced by the tensor SVD [Kilmer and Martin, 2011].\nIt plays a similar role as the matrix nuclear norm which is the convex\nsurrogate of the matrix rank. Considering that the TNN based Tensor Robust PCA\n[Lu et al., 2018a] is an elegant extension of Robust PCA with a similar tight\nrecovery bound, it is natural to solve other low rank tensor recovery problems\nextended from the matrix cases. However, the extensions and proofs are\ngenerally tedious. The general atomic norm provides a unified view of\nlow-complexity structures induced norms, e.g., the $\\ell_1$-norm and nuclear\nnorm. The sharp estimates of the required number of generic measurements for\nexact recovery based on the atomic norm are known in the literature. In this\nwork, with a careful choice of the atomic set, we prove that TNN is a special\natomic norm. Then by computing the Gaussian width of certain cone which is\nnecessary for the sharp estimate, we achieve a simple bound for guaranteed low\ntubal rank tensor recovery from Gaussian measurements. Specifically, we show\nthat by solving a TNN minimization problem, the underlying tensor of size\n$n_1\\times n_2\\times n_3$ with tubal rank $r$ can be exactly recovered when the\ngiven number of Gaussian measurements is $O(r(n_1+n_2-r)n_3)$. It is order\noptimal when comparing with the degrees of freedom $r(n_1+n_2-r)n_3$. Beyond\nthe Gaussian mapping, we also give the recovery guarantee of tensor completion\nbased on the uniform random mapping by TNN minimization. Numerical experiments\nverify our theoretical results.","url_abs":"http://arxiv.org/abs/1806.02511v1","url_pdf":"http://arxiv.org/pdf/1806.02511v1.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":"exact-low-tubal-rank-tensor-recovery-from","repo_url":"https://github.com/canyilu/tensor-completion-tensor-recovery","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[{"method_slug":"pca","method_name":"PCA"}],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":"https://syntology.ai/paper/1806.02511","atlas_url":"https://app.syntology.ai/?focus=1806.02511","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}