General Mathematics


Rip-Based Performance Guarantee for Low Tubal Rank Tensor Recovery

Authors: Feng Zhang, Wendong Wang, Jianwen Huang, Jianjun Wang

The essential task of multi-dimensional data analysis focuses on the tensor decomposition and the corresponding notion of rank. However, most tensor ranks are not well defined with a tight convex relaxation. In this paper, by introducing the notion of tensor singular value decomposition (t-SVD), we establish a regularized tensor nuclear norm minimization (RTNNM) model for low tubal rank tensor recovery. In addition, the tensor nuclear norm within the unit ball of the tensor spectral norm here has been shown to be a convex envelop of tensor average rank. On the other hand, many variants of the restricted isometry property (RIP) have proven to be crucial frameworks and analysis tools for recovery of sparse vectors and low-rank tensors. So, we initiatively define a novel tensor restrict isometry property (t-RIP) based on t-SVD. Besides, our theoretical results show that any third-order tensor X∈R^{n_{1}× n_{2}× n_{3}} whose tubal rank is at most r can stably be recovered from its as few as measurements y = M(X)+w with a bounded noise constraint ||w||_{2}≤ε via the RTNNM model, if the linear map M obeys t-RIP with δ_{tr}^{M}<√(t-1)/(n_{3}^{2}+t-1) for certain fixed t>1. Surprisingly, when n_{3}=1, our conditions coincide with T. Cai and A. Zhang's sharp work in 2013 for low-rank matrix recovery via the constrained nuclear norm minimization. We note that, as far as the authors are aware, such kind of result has not previously been reported in the literature.

Comments: 25 Pages.

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Submission history

[v1] 2018-10-08 22:43:29 (removed)
[v2] 2018-10-14 08:59:07
[v3] 2018-11-22 03:02:24

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