Tensor Tubal Rank Research Articles

Balanced Unfolding Induced Tensor Nuclear Norms for High-Order Tensor Completion.

The recently proposed tensor tubal rank has been witnessed to obtain extraordinary success in real-world tensor data completion. However, existing works usually fix the transform orientation along the third mode and may fail to turn multidimensional low-tubal-rank structure into account. To alleviate these bottlenecks, we introduce two unfolding induced tensor nuclear norms (TNNs) for the tensor completion (TC) problem, which naturally extends tensor tubal rank to high-order data. Specifically, we show how multidimensional low-tubal-rank structure can be captured by utilizing a novel balanced unfolding strategy, upon which two TNNs, namely, overlapped TNN (OTNN) and latent TNN (LTNN), are developed. We also show the immediate relationship between the tubal rank of unfolding tensor and the existing tensor network (TN) rank, e.g., CANDECOMP/PARAFAC (CP) rank, Tucker rank, and tensor ring (TR) rank, to demonstrate its efficiency and practicality. Two efficient TC models are then proposed with theoretical guarantees by analyzing a unified nonasymptotic upper bound. To solve optimization problems, we develop two alternating direction methods of multipliers (ADMM) based algorithms. The proposed models have been demonstrated to exhibit superior performance based on experimental findings involving synthetic and real-world tensors, including facial images, light field images, and video sequences.

A novel $ \ell_{0} $ minimization framework of tensor tubal rank and its multi-dimensional image completion application

A novel $ \ell_{0} $ minimization framework of tensor tubal rank and its multi-dimensional image completion application

Low rank tensor recovery by schatten capped p norm and plug-and-play regularization

Low rank tensor recovery (LRTR) problem has attracted wide attentions and researches, and has been applied to various fields. In this paper, we mainly consider the nonconvex relaxtion methods for LRTR. Firstly, we newly propose the tensor schatten capped p (SCp) norm as a nonconvex surrogate of the tensor tubal rank, which can provide a closer approximation of the tubal rank compared with some existing nonconvex approximations. Secondly, we introduce an implicit Plug-and-Play (PnP)-based regularization into the nonconvex model to further improve the recovered performance. The proposed model is abbreviated as SCp-PnP. Based on the alternating direction method of multipliers, we propose an efficient algorithm to solve SCp-PnP. The convergence of this algorithm is also provided. Finally, extensive experiments on different kinds of datasets show that our method significantly outperforms several state-of-the-art LRTR methods in terms of both quality metrics and visual effects.

Robust low tubal rank tensor completion via factor tensor norm minimization

Recent research has demonstrated that low tubal rank recovery based on tensor has received extensive attention. In this correspondence, we define tensor double nuclear norm and tensor Frobenius/nuclear hybrid norm to induce a surrogate for tensor tubal rank, and prove that they are equivalent to tensor Schatten-p norm for p=1/2 and p=2/3. Based on the definition, we propose two novel tractable tensor completion models called Double Nuclear norm regularized Tensor Completion (DNTC) and Frobenius/Nuclear hybrid norm regularized Tensor Completion (FNTC) by integrating these two norm minimization and factorization methods into a joint learning framework. Furthermore, we adopt invertible linear transforms to obtain low tubal rank tensors, which makes the model more flexible and effective. Two efficient algorithms are designed to solve the proposed tensor completion models by incorporating the convexity of the factor norms. Comprehensive experiments are conducted on synthetic and real datasets to achieve better results in comparison with some state-of-the-art approaches.

A new nonconvex relaxation approach for low tubal rank tensor recovery

In this paper, we consider the tensor recovery problem within the low tubal rank framework. A new nonconvex surrogate is proposed to approximate the tensor tubal rank. The main advantage is that it uses a class of nonconvex functions to penalize the singular tubes directly instead of penalizing the singular values as in existing methods. Our proposed surrogate can continuously approximate the tubal rank without breaking its composition structure and keep the intrinsic structural information of the tensor better than existing methods. We then develop an efficient iteratively reweighted tube thresholding algorithm to solve the tensor recovery model equipped with the new tubal rank surrogate and provide the theoretical guarantee for convergence. Simulation results on synthetical and practical data demonstrate the superior performance of the proposed method over several widely used methods in the literature.

Low tubal rank tensor completion based on singular value factors

Inspired by the definition of tensor-tensor product and tensor tubal rank, a randomized singular value decomposition of tensor is presented in this paper. Based on tensor singular value decomposition (t-SVD) and tensor randomized singular value decomposition (t-RSVD), we obtain two efficient algorithms to solve tensor completion problem. We also propose the adaptive rank method to adjust the tubal rank of tensor. The main advantage of the random projection-based t-RSVD is to cut down the computing time in consideration of large-scale problems. In the optimization process, the alternating minimization algorithm is employed to solve the tensor completion problem. Finally, numerical experiment results indicate that the t-RSVD is competitive and consumes less time than the t-SVD. The efficiency and feasibility of our methods are illustrated by the image and video recovery.

Fast randomized tensor singular value thresholding for low‐rank tensor optimization

AbstractLow‐rank tensor optimization can be converted to a convex optimization problem, which minimizes a convex surrogate to the tensor tubal rank. This problem can be solved iteratively by applying a closed‐form proximal operator, called tensor singular value thresholding (t‐SVT). But they suffer from high computational cost of tensor singular value decomposition (t‐SVD) at each iteration. In this article, we propose a fast randomized algorithm for t‐SVT, which is called as FR t‐SVT. The key idea is to extract an approximate basis for the range of each frontal slice of a third‐order tensor in the Fourier domain from its compressed part. Our theoretical analysis shows the relationship between the approximation bounds of t‐SVD and its effect to the problem via t‐SVT. Numerical examples illustrate the effectiveness of our model.

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>

ST-SVD factorization and s-diagonal tensors

A third order real tensor is mapped to a special f-diagonal tensor by going through Discrete Fourier Transform (DFT), standard matrix SVD and inverse DFT. We call such an f-diagonal tensor an s-diagonal tensor. An f-diagonal tensor is an s-diagonal tensor if and only if it is mapped to itself in the above process. The third order tensor space is partitioned to orthogonal equivalence classes. Each orthogonal equivalence class has a unique s-diagonal tensor. Two s-diagonal tensors are equal if they are orthogonally equivalent. Third order tensors in an orthogonal equivalence class have the same tensor tubal rank and T-singular values. Four meaningful necessary conditions for s-diagonal tensors are presented. Then we present a set of sufficient and necessary conditions for s-diagonal tensors. Such conditions involve a special complex number. In the cases that the dimension of the third mode of the considered tensor is $2, 3$ and $4$, we present direct sufficient and necessary conditions which do not involve such a complex number.

Infrared dim target detection via mode-k1k2 extension tensor tubal rank under complex ocean environment

Infrared dim target detection under complex ocean environment plays a key role in military and civilian fields. Many state-of-the-art methods have disadvantages such as low generalization ability, poor robustness to noise and stubborn edges, high time complexity, and the existence of background residuals or target defects in detection results. To further solve these shortcomings, based on the infrared patch tensor (IPT) model, a robust infrared dim target detection algorithm is proposed in this paper, which converts the target detection task into a convex optimization problem. Aiming at the current situation that the tensor rank approximation in the IPT model has not been well resolved, a new vector form of tensor rank named mode-k1k2 extension tensor tubal rank (METTR) is defined, the elements of which include the tubal ranks of all tensors expanded via mode-k1k2 extension. Through the mode-k1k2 extension of the tensor, the hidden information among the different modes of the tensor is better mined. To minimize the METTR efficiently, we propose its convex approximation norm METTR, and establish a tensor robust principal component analysis (TRPCA) model joint l1 norm. Then we use the alternating direction multiplier method (ADMM) and set the optimal parameters to solve the proposed model. A series of experimental results show that the proposed algorithm outperforms the baselines in terms of background suppression and target detection.

Low‐rank tensor completion with sparse regularization in a transformed domain

AbstractTensor completion is a challenging problem with various applications, especially in recovering incomplete visual data. Considering a color image or gray video as a three‐dimensional tensor, many related models based on the low‐rank prior of the tensor have been proposed. However, the low‐rank prior may not be enough to recover the original tensor from the observed incomplete tensor. In this paper, we propose a tensor completion method to recover color images and gray videos by exploiting both the low‐rank and sparse prior of the observed tensor. Specifically, the tensor completion task can be formulated as a low‐rank minimization problem with a sparse regularizer. The low‐rank property is depicted by the tensor truncated nuclear norm based on tensor singular value decomposition which is a better approximation of tensor tubal rank than tensor nuclear norm. While the sparse regularizer is imposed by a ℓ1‐norm in a discrete cosine transformation domain, which can better employ the local sparse property of the incomplete data. To solve the optimization problem, we employ an alternating direction method of multipliers in which we only need to solve several subproblems which have closed‐form solutions. Substantial experiments on real‐world images and videos show that the proposed method has better performances than the existing state‐of‐the‐art methods.

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.

Tensor N-tubal rank and its convex relaxation for low-rank tensor recovery

The recent popular tensor tubal rank, defined based on tensor singular value decomposition (t-SVD), yields promising results. However, its framework is applicable only to three-way tensors and lacks the flexibility necessary tohandle different correlations along different modes. To tackle these two issues, we define a new tensor unfolding operator, named mode-k1k2 tensor unfolding, as the process of lexicographically stacking all mode-k1k2 slices of an N-way tensor into a three-way tensor, which is a three-way extension of the well-known mode-k tensor matricization. On this basis, we define a novel tensor rank, named the tensor N-tubal rank, as a vector consisting of the tubal ranks of all mode-k1k2 unfolding tensors, to depict the correlations along different modes. To efficiently minimize the proposed N-tubal rank, we establish its convex relaxation: the weighted sum of the tensor nuclear norm (WSTNN). Then, we apply the WSTNN to low-rank tensor completion (LRTC) and tensor robust principal component analysis (TRPCA). The corresponding WSTNN-based LRTC and TRPCA models are proposed, and two efficient alternating direction method of multipliers (ADMM)-based algorithms are developed to solve the proposed models. Numerical experiments demonstrate that the proposed models significantly outperform the compared ones.

Weighted t-Schatten-p Norm Minimization for Real Color Image Denoising

In this paper, to fully exploit the spatial and spectral correlation information, we present a new real color image denoising scheme using tensor Schatten-p norm (t-Schatten-p norm) minimization based on t-SVD to recover the underlying low-rank tensor. Similar to matrix Schatten-p norm, using non-convex t-Schatten-p (0 <; p <; 1) norm minimization could obtain better results than the tensor nuclear norm minimization which is a convex relaxation of the nonconvex tensor tubal rank. To avoid over-shrink the tensor tubal rank components, a flexible weighted t-Schatten-p norm model is proposed with weights assigned to different elements of tensor singular tubes. We adopt the generalized iterated shrinkage algorithm to solve the minimization problem efficiently. Extensive experiments on one synthetic and two realistic datasets demonstrate the effectiveness of our proposed method to remove noise both quantitatively and qualitatively.

Simultaneous Reconstruction and Moving Object Detection From Compressive Sampled Surveillance Videos

The spatially distributed digital cameras in Wireless Multimedia Sensor Networks (WMSN) are provided with miniature batteries resulting in power constraints. These cameras acquire compressive measurements of video at a rate significantly below the Nyquist rate and transmit them wirelessly in the network. Thus, the encoding side (transmitter) is made less complex at the expense of increased complexity at the more resourceful decoder (receiver). Well grounded on this relevant practical scenario, a unified system is proposed that integrates detection of moving objects into the Compressive Sensing (CS) recovery framework and thereby realizing simultaneous data recovery and object detection in a single optimization problem which guarantees fast response. In this work, a new tensor RPCA approach is proposed to accomplish this requirement. For background separation, low rank approximation is done on the highly correlated background components. A Laplace function based surrogate for tensor tubal rank is formulated to provide adaptive thresholding for the singular value tubes of the background tensor. Moreover, the spatio-temporal continuity of the foreground is explored using 3D-Piecewise Smoothness Constraints combinations based Anisotropic Total Variation (3D-PSCATV) regularization. Additionally, $l_{1}$ regularization has been adopted to describe the sparsity of moving objects. The proposed model is solved using Alternative Direction Method of Multipliers (ADMM) scheme. The quantitative and qualitative results validate the superior performance of the proposed method against the compared approaches.

Uniqueness Guarantee of Solutions of Tensor Tubal-Rank Minimization Problem

This letter considers the recovery of a low-tubal-rank tensor from incomplete linear observations. It is shown that the unknown tensor $\boldsymbol{\mathcal {Z}}\in \mathbb {R}^{n_{1} \times n_{2} \times n_{3}}$ of tubal-rank $r$ can be reconstructed as a unique solution of a tractable method — tensor nuclear norm (TNN) minimization, provided that the number of Gaussian observations $m\geq 3r(n_{1} + n_{2} - r)n_{3}+1$ . In this work, we examine the fundamental question of the minimal number of linear observations needed to reconstruct the tensor $\boldsymbol{\mathcal {Z}}$ from these observations, regardless of the practicality of the reconstruction scheme. Consequently, we provide two benchmark results so that different reconstruction schemes including TNN minimization can be compared to each other. Specifically, we conclude that $m\geq 2r(n_{1} + n_{2} - 2r)n_{3}$ and $m\geq r(n_{1} + n_{2} - r)n_{3}+1$ Gaussian observations are necessary and sufficient to guarantee uniform recovery and nonuniform recovery using tensor tubal-rank minimization method, respectively.

Mixed Noise Removal in Hyperspectral Image via Low-Fibered-Rank Regularization

The tensor tubal rank, defined based on the tensor singular value decomposition (t-SVD), has obtained promising results in hyperspectral image (HSI) denoising. However, the framework of the t-SVD lacks flexibility for handling different correlations along different modes of HSIs, leading to suboptimal denoising performance. This article mainly makes three contributions. First, we introduce a new tensor rank named tensor fibered rank by generalizing the t-SVD to the mode-k t-SVD, to achieve a more flexible and accurate HSI characterization. Since directly minimizing the fibered rank is NP-hard, we suggest a three-directional tensor nuclear norm (3DTNN) and a three-directional log-based tensor nuclear norm (3DLogTNN) as its convex and nonconvex relaxation to provide an efficient numerical solution, respectively. Second, we propose a fibered rank minimization model for HSI mixed noise removal, in which the underlying HSI is modeled as a low-fibered-rank component. Third, we develop an efficient alternating direction method of multipliers (ADMMs)-based algorithm to solve the proposed model, especially, each subproblem within ADMM is proven to have a closed-form solution, although 3DLogTNN is nonconvex. Extensive experimental results demonstrate that the proposed method has superior denoising performance, as compared with the state-of-the-art competing methods on low-rank matrix/tensor approximation and noise modeling.

Real Color Image Denoising Using t-Product- Based Weighted Tensor Nuclear Norm Minimization

Color images can be seen as third-order tensors with column, row and color modes. Considering two inherent characteristics of a color image including the non-local self-similarity (NSS) and the cross-channel correlation, we extract non-local similar patch groups from a color image and treat these groups as tensors with each color channel corresponding to the frontal slice of the tensor to exploit the information within and cross channel correlation. Inspired by recently proposed tensor-tensor product (t-product), t-SVD, tensor tubal rank and rigorously deduced tensor nuclear norm, a novel t-product based weighted tensor nuclear norm minimization (WTNNM) is proposed to model the extracted non-local similar patch group tensor (NPGT). Considering the NPGT is of low tubal rank, we formulate real color image denoising as a low tubal rank tensor recovery problem and solve it with the weighted tensor nuclear norm minimization. Experiments on both simulated and realistic noisy images verify the effectiveness of our method.

Nonconvex Low Tubal Rank Tensor Minimization

In the sparse vector recovery problem, the L 0 -norm can be approximated by a convex function or a nonconvex function to achieve sparse solutions. In the low-rank matrix recovery problem, the nonconvex matrix rank can be replaced by a convex function or a nonconvex function on the singular value of matrix to achieve low-rank solutions. Although the convex relaxation can easily lead to the optimal solution, the nonconvex approximation tends to yield more sparse or lower rank local solutions. As a natural extension of vector and matrix to high order structure, tensor can better represent the essential structure of data for modeling the high-dimensional data. In this paper, we study the low tubal rank tensor recovery problem by nonconvex optimization. Instead of using convex tensor nuclear norm, we use nonconvex surrogate functions to approximate the tensor tubal rank, and propose a tensor based iteratively reweighted nuclear norm solver. We further provide the convergence analysis of our new solver. Sufficient experiments on synthetic data and real images verify the effectiveness of our new method.

Orthogonal tubal rank-1 tensor pursuit for tensor completion

This work addresses the issue of tensor completion. The properties of the tensor tubal rank are firstly discussed. It is shown that the tensor tubal rank has similar properties like that of matrix rank derived from SVD. The completion algorithm for the case that the measurements are noise-free or corrupted by Gaussian noise is then proposed based on an orthogonal pursuit on tubal rank-1 tensors. The philosophy behind the devised approach is to relax the problem of tensor tubal rank minimization into tensor Frobenius-norm optimization with a constraint on the maximum number of orthogonal tensors. An iterative procedure which calculates one orthogonal tensor at each iterative step is then suggested, and the local convergence under the noise-free case is also proved. Furthermore, the proposed method is generalized to the situation where the observations are corrupted by impulsive noise in a tubal form. To tackle the impulsive noise, we formulate the problem of tensor completion as minimization of tensor tubal ℓp-norm with 1 < p < 2. An iteratively reweighted procedure is employed to compute the orthogonal tensors. The algorithms are compared with the state-of-the-art approaches using both synthetic data and real data sets.