{"type":"ScholarlyArticle","title":"Exact Low Tubal Rank Tensor Recovery from Gaussian Measurements","authors":[{"name":"Canyi Lu","slug":"canyi-lu","given":"Canyi","family":"Lu","affiliation":"Department of Electrical and Computer Engineering , Carnegie Mellon University"},{"name":"Jiashi Feng","slug":"jiashi-feng","given":"Jiashi","family":"Feng","affiliation":"Department of Electrical and Computer Engineering , National University of Singapore"},{"name":"Zhouchen Lin","slug":"zhouchen-lin","given":"Zhouchen","family":"Lin","affiliation":"Key Laboratory of Machine Perception (MOE) , School of EECS , Peking University; Cooperative Medianet Innovation Center , Shanghai Jiao Tong University"},{"name":"Shuicheng Yan","slug":"shuicheng-yan","given":"Shuicheng","family":"Yan","affiliation":"360 AI Institute; Department of Electrical and Computer Engineering , National University of Singapore"}],"abstract":"The recent proposed Tensor Nuclear Norm (TNN) [Lu et al., 2016; 2018a] is an interesting convex penalty induced by the tensor SVD [Kilmer and Martin, 2011]. It plays a similar role as the matrix nuclear norm which is the convex surrogate of the matrix rank. Considering that the TNN based Tensor Robust PCA [Lu et al., 2018a] is an elegant extension of Robust PCA with a similar tight recovery bound, it is natural to solve other low rank tensor recovery problems extended from the matrix cases. However, the extensions and proofs are generally tedious. The general atomic norm provides a unified view of low-complexity structures induced norms, e.g., the l1-norm and nuclear norm. The sharp estimates of the required number of generic measurements for exact recovery based on the atomic norm are known in the literature. In this work, with a careful choice of the atomic set, we prove that TNN is a special atomic norm. Then by computing the Gaussian width of certain cone which is necessary for the sharp estimate, we achieve a simple bound for guaranteed low tubal rank tensor recovery from  Gaussian measurements. Specifically, we show that by solving a TNN minimization problem, the underlying tensor of size n1×n2×n3 with tubal rank r can be exactly recovered when the given number of Gaussian measurements is O(r(n1+n2−r)n3). It is order optimal when comparing with the degrees of freedom r(n1+n2−r)n3. Beyond the Gaussian mapping, we also give the recovery guarantee of tensor completion based on the uniform random mapping by TNN minimization. Numerical experiments verify our theoretical results.","doi":"10.24963/ijcai.2018/347","url":"https://ijcai.org/proceedings/2018/347","pdf":"https://ijcai.org/proceedings/2018/0347.pdf","markdown":"https://ijcai.org/proceedings/2018/347.md","bibtex":"https://ijcai.org/proceedings/2018/bibtex/347","volume":{"slug":"2018","title":"Proceedings of the Twenty-Seventh International Joint Conference on Artificial Intelligence","short_title":"IJCAI 2018","url":"https://ijcai.org/proceedings/2018/","year":2018,"location":"Stockholm","dates":"13-19 July 2018","editors":"Jérôme Lang","isbn":"978-0-9992411-2-7","publisher":"International Joint Conferences on Artificial Intelligence Organization","doi_prefix":"10.24963/ijcai.2018"},"section":"Machine Learning","track":"Main track","number":347,"pages":{"first":2504,"last":2510},"keywords":["Machine Learning: Feature Selection ; Learning Sparse Models","Computer Vision: Statistical Methods and Machine Learning"],"videos":[],"references":[{"position":1,"raw":"[Amelunxen et al., 2014] Dennis Amelunxen, Martin Lotz, Michael B McCoy, and Joel A Tropp. Living on the edge: Phase transitions in convex programs with random data. 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