RIP-based performance guarantee for low-tubal-rank tensor recovery
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The essential task of multi-dimensional data analysis focuses on the tensor decomposition and the corresponding notion of rank. 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. 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 restricted 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 Cai and 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.
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