Tensor Recovery Research Articles

AG-LRTR: An Adaptive and Generic Low-Rank Tensor-Based Recovery for IIoT Network Traffic Factors Denoising

The ubiquitous 5G-enable industrial Internet of Things interconnects a great number of intelligent sensors and actors. Network management becomes challenging due to massive traffic data generated by industrial equipment. However, the conventional single traffic factor is insufficient for the increasingly complicated network engineering tasks due to the poor representation capability. Besides, the insecure equipment with open communication access easily brings irregular network fluctuations to network traffic which interferes with the primary traffic factor. The simple and interfered traffic factor decreases the network management efficiency and misleads the operators. Motivated by that, we construct a comprehensive tensor model representing multi-dimension traffic factors to describe the network traffic beneficial characteristics. Meanwhile, an adaptive and generic low-rank tensor recovery (AG-LRTR) algorithm in the tensor singular value decomposition (t-SVD) framework is proposed for denoising. For effective tensor recovery, the alternating direction method of multipliers is employed to theoretically solve the partial augmented Lagrangian function of our objective with a closed-form solution. Numerical experiments on both synthetic data and real-world traffic data in IIoT validate that our proposed algorithm outperforms other state-of-the-art of tensor recovery algorithms.

Infrared small target detection based on dehazing enhancement and tensor recovery

Infrared small target detection based on dehazing enhancement and tensor recovery

ORTP: A Video SAR Imaging Algorithm Based on Low-Tubal-Rank Tensor Recovery

ORTP: A Video SAR Imaging Algorithm Based on Low-Tubal-Rank Tensor Recovery

Joint Low-Rank and Sparse Tensors Recovery for Video Synthetic Aperture Radar Imaging

Video synthetic aperture radar (SAR) receives more and more attention in recent years because it can provide continuous images of the observed scene. However, the enormous data of video SAR to obtain the multiframe images bring big challenges to its transmission, storage, and processing, especially for small unmanned aerial vehicle (UAV) platform. In this article, we aim at proposing an efficient video formation method for video SAR systems with reduced data. First, the characteristics of video SAR observed scene are analyzed. It is found that the observed scene with multiple frames can be modeled as the sum of a low-rank tensor and a sparse tensor efficiently. After that, the video formation problem for video SAR is modeled as a joint low-rank and sparse tensors recovery problem. Finally, an efficient tensor alternating direction method of multiplier is proposed to obtain the final SAR video. Compared with the traditional frequency- or time-domain imaging methods, the amount of data samples can be greatly reduced. On the other hand, the proposed method outperforms the state-of-the-art SAR imaging methods with reduced samples, including the joint low-rank and sparse matrices recovery method and the low-rank tensor recovery method. Numerical simulations validate the effectiveness of the proposed method.

An orthogonal equivalence theorem for third order tensors

<p style='text-indent:20px;'>In 2011, Kilmer and Martin proposed tensor singular value decomposition (T-SVD) for third order tensors. Since then, T-SVD has applications in low rank tensor approximation, tensor recovery, multi-view clustering, multi-view feature extraction, tensor sketching, etc. By going through the Discrete Fourier Transform (DFT), matrix SVD and inverse DFT, a third order tensor is mapped to an f-diagonal third order tensor. We call this a Kilmer-Martin mapping. We show that the Kilmer-Martin mapping of a third order tensor is invariant if that third order tensor is taking T-product with some orthogonal tensors. We define singular values and T-rank of that third order tensor based upon its Kilmer-Martin mapping. Thus, tensor tubal rank, T-rank, singular values and T-singular values of a third order tensor are invariant when it is taking T-product with some orthogonal tensors. Some properties of singular values, T-rank and best T-rank one approximation are discussed.</p>

Efficient Tensor Robust PCA Under Hybrid Model of Tucker and Tensor Train

Tensor robust principal component analysis (TRPCA) is a fundamental model in machine learning and computer vision. Recently, tensor train (TT) decomposition has been verified effective to capture the global low-rank correlation for tensor recovery tasks. However, due to the large-scale tensor data in real-world applications, existing TRPCA models often suffer from high computational complexity. In this letter, we propose an efficient TRPCA under hybrid model of Tucker and TT. Specifically, in theory we reveal that TT nuclear norm (TTNN) of the original big tensor can be equivalently converted to that of a much smaller tensor via a Tucker compression format, thereby significantly reducing the computational cost of singular value decomposition (SVD). Numerical experiments on both synthetic and real-world tensor data verify the superiority of the proposed model.

Hyperspectral Image Restoration Using 3-D Hybrid Higher Degree Total Variation Regularized Nonconvex Local Low-Rank Tensor Recovery

The degradation of spaceborne hyperspectral images (HSIs) usually results from various types of noise. In this letter, we propose a 3D hybrid higher degree total variation regularized nonconvex local low-rank tensor recovery (H <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> DTV-NLRTR) model to restore the HSIs. Inspired by the good performance of the higher degree total variation penalty in image denoising, we first develop a 3D hybrid higher degree total variation penalty term, which is able to capture the fine image details and edges along the spatial dimensions and spectral dimension. The tensor multi-Schatten- <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</i> norm is chosen as the relaxation of the low-rank tensor constraint, which can not only separate the low-rank clean HSI patches from noisy images effectively but also improve the computational efficiency. The proposed H <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> DTV-NLRTR model can simultaneously characterize the spectral correlation and the spatial structure of the HSI dataset by incorporating the H <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> DTV penalty in the nonconvex local low-rank tensor recovery problem. In addition, we adopt a fast iterative majorize-minimize algorithm to efficiently solve the corresponding optimization problem. The numerical experiments on both simulated and real HSI datasets demonstrate that the proposed algorithm provides consistently improved restoration results compared with the state-of-the-art algorithms.

Low-Rank High-Order Tensor Completion With Applications in Visual Data.

Recently, tensor Singular Value Decomposition (t-SVD)-based low-rank tensor completion (LRTC) has achieved unprecedented success in addressing various pattern analysis issues. However, existing studies mostly focus on third-order tensors while order- d ( d ≥ 4 ) tensors are commonly encountered in real-world applications, like fourth-order color videos, fourth-order hyper-spectral videos, fifth-order light-field images, and sixth-order bidirectional texture functions. Aiming at addressing this critical issue, this paper establishes an order- d tensor recovery framework including the model, algorithm and theories by innovatively developing a novel algebraic foundation for order- d t-SVD, thereby achieving exact completion for any order- d low t-SVD rank tensors with missing values with an overwhelming probability. Emperical studies on synthetic data and real-world visual data illustrate that compared with other state-of-the-art recovery frameworks, the proposed one achieves highly competitive performance in terms of both qualitative and quantitative metrics. In particular, as the observed data density becomes low, i.e., about 10%, the proposed recovery framework is still significantly better than its peers. The code of our algorithm is released at https://github.com/Qinwenjinswu/TIP-Code.

Iterative hard thresholding for low CP-rank tensor models

Recovery of low-rank matrices from a small number of linear measurements is now well-known to be possible under various model assumptions on the measurements. Such results demonstrate robustness and are backed with provable theoretical guarantees. However, extensions to tensor recovery have only recently began to be studied and developed, despite an abundance of practical tensor applications. Recently, a tensor variant of the Iterative Hard Thresholding method was proposed and theoretical results were obtained that exact guarantee recovery of tensors with low Tucker rank. In this paper, we utilize and prove a similar tensor version of the Restricted Isometry Property (RIP) to extend these results for tensors with low CANDECOMP/PARAFAC (CP) rank. In doing so, we leverage recent results on efficient approximations of CP decompositions that remove the need for challenging assumptions in prior works. We complement our theoretical findings with empirical results that showcase the potential of the approach.

Tensor Q-rank: new data dependent definition of tensor rank

Recently, the \({ Tensor}~{ Nuclear}~{ Norm}~{ (TNN)}\) regularization based on t-SVD has been widely used in various low tubal-rank tensor recovery tasks. However, these models usually require smooth change of data along the third dimension to ensure their low rank structures. In this paper, we propose a new definition of data dependent tensor rank named tensor Q-rank by a learnable orthogonal matrix \(\mathbf {Q}\), and further introduce a unified data dependent low rank tensor recovery model. According to the low rank hypothesis, we introduce two explainable selection methods of \(\mathbf {Q}\), under which the data tensor may have a more significant low tensor Q-rank structure than that of low tubal-rank structure. Specifically, maximizing the variance of singular value distribution leads to Variance Maximization Tensor Q-Nuclear norm (VMTQN), while minimizing the value of nuclear norm through manifold optimization leads to Manifold Optimization Tensor Q-Nuclear norm (MOTQN). Moreover, we apply these two models to the low rank tensor completion problem, and then give an effective algorithm and briefly analyze why our method works better than TNN based methods in the case of complex data with low sampling rate. Finally, experimental results on real-world datasets demonstrate the superiority of our proposed models in the tensor completion problem with respect to other tensor rank regularization models.

Fast Tensor Nuclear Norm for Structured Low-Rank Visual Inpainting

Low-rank modeling has achieved great success in visual data completion. However, the low-rank assumption of original visual data may be in approximate mode, which leads to suboptimality for the recovery of underlying details, especially when the missing rate is extremely high. In this paper, we go further by providing a detailed analysis about the rank distributions in Hankel structured and clustered cases, and figure out both non-local similarity and patch-based structuralization play a positive role. This motivates us to develop a new Hankel low-rank tensor recovery method that is competent to truthfully capture the underlying details with sacrifice of slightly more computational burden. First, benefiting from the correlation of different spectral bands and the smoothness of local spatial neighborhood, we divide the visual data into overlapping 3D patches and group the similar ones into individual clusters exploring the non-local similarity. Second, the 3D patches are individually mapped to the structured Hankel tensors for better revealing low-rank property of the image. Finally, we solve the tensor completion model via the well-known alternating direction method of multiplier (ADMM) optimization algorithm. Due to the fact that size expansion happens inevitably in Hankelization operation, we further propose a fast randomized skinny tensor singular value decomposition (rst-SVD) to accelerate the per-iteration running efficiency. Extensive experimental results on real world datasets verify the superiority of our method compared to the state-of-the-art visual inpainting approaches.

On the Synergy between Nonconvex Extensions of the Tensor Nuclear Norm for Tensor Recovery

Low-rank tensor recovery has attracted much attention among various tensor recovery approaches. A tensor rank has several definitions, unlike the matrix rank—e.g., the CP rank and the Tucker rank. Many low-rank tensor recovery methods are focused on the Tucker rank. Since the Tucker rank is nonconvex and discontinuous, many relaxations of the Tucker rank have been proposed, e.g., the sum of nuclear norm, weighted tensor nuclear norm, and weighted tensor schatten-p norm. In particular, the weighted tensor schatten-p norm has two parameters, the weight and p, and the sum of nuclear norm and weighted tensor nuclear norm are special cases of these parameters. However, there has been no detailed discussion of whether the effects of the weighting and p are synergistic. In this paper, we propose a novel low-rank tensor completion model using the weighted tensor schatten-p norm to reveal the relationships between the weight and p. To clarify whether complex methods such as the weighted tensor schatten-p norm are necessary, we compare them with a simple method using rank-constrained minimization. It was found that the simple methods did not outperform the complex methods unless the rank of the original tensor could be accurately known. If we can obtain the ideal weight, p=1 is sufficient, although it is necessary to set p&lt;1 when using the weights obtained from observations. These results are consistent with existing reports.

Tensor Recovery via $*_L$-Spectral $k$-Support Norm

Unlike traditional tensor decompositions which model low-rankness in the original domain, the recently proposed tensor <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">*L</sub> -Singular Value Decomposition ( <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">*L</sub> -SVD) casts a new light on tensor analysis by exploiting low-rankness in the spectral domain. For convexized rank minimization within the framework of <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">*L</sub> -SVD, the tensor <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">*L</sub> -Tubal Nuclear Norm ( <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">*L</sub> -TNN) out-performs classical tensorial extensions of matrix nuclear norm in many applications. In this paper, we first generalize <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">*L</sub> -TNN to the <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> *L</sub> -Spectral k-Support Norm ( <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">*L</sub> -SpSN-k)as a new low-rank regularizer, and then adopt it to formulate two estimators for tensor recovery from noisy linear observations. Further, statistical performance of the proposed estimators is analyzed by establishing both deterministic and non-asymptotic upper bounds on the estimation error, which indicates that the proposed tensor Dantzig selector enjoys near-optimal estimation error for tensor compressive sensing and tensor completion. Moreover, proximal operator of the proposed norm is derived and embeded in an efficient algorithm to compute the estimators. Experiments on synthetic datasets verify correctness of the proposed error bounds and image inpainting results demonstrate superiority of the proposed estimators.

Generalized Nonconvex Approach for Low-Tubal-Rank Tensor Recovery.

The tensor-tensor product-induced tensor nuclear norm (t-TNN) (Lu et al., 2020) minimization for low-tubal-rank tensor recovery attracts broad attention recently. However, minimizing the t-TNN faces some drawbacks. For example, the obtained solution could be suboptimal to the original problem due to its loose approximation. In this article, we extract a unified nonconvex surrogate of the tensor tubal rank as a tighter regularizer, which involves many popular nonconvex penalty functions. An iterative reweighted t-TNN algorithm is proposed to solve the resulting generalized nonconvex tubal rank minimization for tensor recovery. It converges to a critical point globally with rigorous proofs based on the Kurdyka-Łojasiwicz property. Furthermore, we provide the theoretical guarantees for exact and robust recovery by developing the tensor null space property. Extensive experiments demonstrate that our approach markedly enhances recovery performance compared with several state-of-the-art convex and nonconvex methods.

Nonlocal robust tensor recovery with nonconvex regularization * *The research of Bai was supported in part by the National Natural Science Foundation of China under Grant No. 11971159. The research of Ng was supported in part by the HKRGC GRF 12306616, 12200317, 12300218, 12300519 and 17201020, and HKU Grant No. 104005583. The research of Zhang was supported in part by

The robust tensor recovery problem consists in reconstructing a tensor from a sample of entries corrupted by noise, which has attracted great interest in a wide range of practical situations such as image processing and computer vision. In this paper, we study robust tensor recovery for third-order tensors with different degradations, which aims to recover a tensor from partial observations corrupted by Gaussian noise and sparse noise simultaneously. In contrast to traditional approaches based on the tensor nuclear norm penalty for the low-rank component and the tensor ℓ 1 norm penalty for the sparse component, we propose a nonlocal robust low-rank tensor recovery model with nonconvex regularization (NRTRM) to explore the global low-rankness and nonlocal self-similarity of the underlying tensor. The NRTRM method is first to extract similar patched-tubes to form a third-order sub-tensor. Then a class of nonconvex low-rank penalties and nonconvex sparse penalties are employed to explore the low-rank component and the sparse corruptions for such sub-tensor, respectively. Moreover, a proximal alternating linearized minimization algorithm is developed to solve the resulting model in each group and its convergence is established under very mild conditions. Extensive numerical experiments on both multispectral images and video datasets demonstrate the superior performance of NRTRM in comparison with several state-of-the-art methods.

A tensor rank theory and maximum full rank subtensors

A matrix always has a full rank submatrix such that the rank of this matrix is equal to the rank of that submatrix. This property is one of the corner stones of the matrix rank theory. We call this property the max-full-rank-submatrix property. Tensor ranks play a crucial role in low rank tensor approximation, tensor completion and tensor recovery. However, their theory is still not matured yet. Can we set an axiom system for tensor ranks? Can we extend the max-full-rank-submatrix property to tensors? We explore these in this paper. We first propose some axioms for tensor rank functions. Then we introduce proper tensor rank functions. The CP rank is a tensor rank function, but is not proper. There are two proper tensor rank functions, the max-Tucker rank and the submax-Tucker rank, which are associated with the Tucker decomposition. We define a partial order among tensor rank functions and show that there exists a unique smallest tensor rank function. We introduce the full rank tensor concept, and define the max-full-rank-subtensor property. We show the max-Tucker tensor rank function and the smallest tensor rank function have this property. We define the closure for an arbitrary proper tensor rank function, and show that it is still a proper tensor rank function and has the max-full-rank-subtensor property. An application of the submax-Tucker rank is also presented.

Regularized Kaczmarz Algorithms for Tensor Recovery

Tensor recovery has recently arisen in a lot of application fields, such as transportation, medical imaging, and remote sensing. Under the assumption that signals possess sparse and/or low-rank str...

Robust Low-tubal-rank Tensor Recovery from Binary Measurements

Low-rank tensor recovery (LRTR) is a natural extension of low-rank matrix recovery (LRMR) to high-dimensional arrays, which aims to reconstruct an underlying tensor X from incomplete linear measurements [Formula: see text]. However, LRTR ignores the error caused by quantization, limiting its application when the quantization is low-level. In this work, we take into account the impact of extreme quantization and suppose the quantizer degrades into a comparator that only acquires the signs of [Formula: see text]. We still hope to recover X from these binary measurements. Under the tensor Singular Value Decomposition (t-SVD) framework, two recovery methods are proposed-the first is a tensor hard singular tube thresholding method; the second is a constrained tensor nuclear norm minimization method. These methods can recover a real n1×n2×n3 tensor X with tubal rank r from m random Gaussian binary measurements with errors decaying at a polynomial speed of the oversampling factor λ:=m/((n1+n2)n3r). To improve the convergence rate, we develop a new quantization scheme under which the convergence rate can be accelerated to an exponential function of λ. Numerical experiments verify our results, and the applications to real-world data demonstrate the promising performance of the proposed methods.

Low-rank tensor recovery for Jacobian-based Volterra identification of parallel Wiener-Hammerstein systems

Abstract We consider the problem of identifying a parallel Wiener-Hammerstein structure from Volterra kernels. Methods based on Volterra kernels typically resort to coupled tensor decompositions of the kernels. However, in the case of parallel Wiener-Hammerstein systems, such methods require nontrivial constraints on the factors of the decompositions. In this paper, we propose an entirely different approach: by using special sampling (operating) points for the Jacobian of the nonlinear map from past inputs to the output, we can show that the Jacobian matrix becomes a linear projection of a tensor whose rank is equal to the number of branches. This representation allows us to solve the identification problem as a tensor recovery problem.

Triple Decomposition and Tensor Recovery of Third Order Tensors

Motivated by the Tucker decomposition, in this paper we introduce a new tensor decomposition for third order tensors, which decomposes a third order tensor to three third order factor tensors. Each...