
Tensor Robust Principal Component Analysis with A New Tensor Nuclear Norm
In this paper, we consider the Tensor Robust Principal Component Analysi...
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RIPbased performance guarantee for lowtubalrank tensor recovery
The essential task of multidimensional data analysis focuses on the ten...
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Exact Recovery of Tensor Robust Principal Component Analysis under Linear Transforms
This work studies the Tensor Robust Principal Component Analysis (TRPCA)...
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Relaxed Leverage Sampling for Lowrank Matrix Completion
We consider the problem of exact recovery of any m× n matrix of rank ϱ f...
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Square Deal: Lower Bounds and Improved Relaxations for Tensor Recovery
Recovering a lowrank tensor from incomplete information is a recurring ...
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Interpolating Convex and NonConvex Tensor Decompositions via the Subspace Norm
We consider the problem of recovering a lowrank tensor from its noisy o...
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HighDimensional Dynamic Systems Identification with Additional Constraints
This note presents a unified analysis of the identification of dynamical...
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Exact Low Tubal Rank Tensor Recovery from Gaussian Measurements
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 lowcomplexity structures induced norms, e.g., the ℓ_1norm 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 n_1× n_2× n_3 with tubal rank r can be exactly recovered when the given number of Gaussian measurements is O(r(n_1+n_2r)n_3). It is order optimal when comparing with the degrees of freedom r(n_1+n_2r)n_3. 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.
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