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Recently, a regularized Kaczmarz method has been proposed to solve tensor recovery problems. In this article, we propose a sampling greedy average regularized Kaczmarz method. This method can be viewed as a block or mini‐batch version of the regularized Kaczmarz method, which is based on averaging several regularized Kaczmarz steps with a constant or adaptive extrapolated step size. Also, it is equipped with a sampling greedy strategy to select the working tensor slices from the sensing tensor. We prove that our new method converges linearly in expectation and show that the sampling greedy strategy can exhibit an accelerated convergence rate compared to the random sampling strategy. Numerical experiments are carried out to show the feasibility and efficiency of our new method on various signal/image recovery problems, including sparse signal recovery, image inpainting, and image deconvolution. [ABSTRACT FROM AUTHOR]

Motion capture (MoCap) technology is an essential tool for recording and analyzing movements of objects or humans. However, MoCap systems frequently encounter the challenge of missing data, stemming from mismatched markers, occlusion, or equipment limitations. Recovery of these missing data is imperative to maintain the reliability and integrity of MoCap recordings. This paper introduces a novel application of the tensor framework for MoCap data completion. We propose three completion algorithms based on the canonical polyadic (CP) decomposition of tensors. The first algorithm utilizes CP decomposition to capture the low-rank structure of the tensor. However, relying only on low-rank assumptions may be insufficient to deal with complex motion data. Thus, we propose two modified CP decompositions that incorporate additional information, SmoothCP and SparseCP decompositions. SmoothCP integrates piecewise smoothness prior, while SparseCP incorporates sparsity prior, each aiming to improve the accuracy and robustness of MoCap data recovery. To compare and evaluate the merit of the proposed algorithms over other tensor completion methods in terms of several evaluation metrics, we conduct numerical experiments with different MoCap sequences from the CMU motion capture dataset. [ABSTRACT FROM AUTHOR]

The problem of nonconvex robust tensor principal component analysis consists of recovering the low-rank and sparse part from a tensor corrupted by noise, which attracts a great deal of attention in a wide range of practical situations. However, existing nonconvex methods face a number of problems, the two most important of which are the restrictions on specific nonconvex functions and the information loss in low-rank part. In this paper, we propose a generalized nonconvex robust tensor principal component analysis model that includes some of the most popular nonconvex functions. Furthermore, we propose a partial weighted tensor nuclear norm and the corresponding partial weighted singular value thresholding operator to improve the generalized model, further reducing the loss of underlying tensor information while recovering the low rank tensor. Besides, two ADMM-based nonconvex algorithms are proposed to the generalized nonconvex model and the improved model respectively. We also analyze the convergence of the algorithms, the computational complexity, and the theoretical guarantee of the proposed models. Numerical experimental results on image data and video data show that our proposed models has superior performance compared to several state-of-the-art methods. [ABSTRACT FROM AUTHOR]

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In this paper, we consider the three-order tensor recovery problem within the tensor tubal rank framework. Most of the recent studies under this framework can not handle the tubal rank directly due to its numerical challenges, while approximating it using various surrogate functions. Although the optimization models derived from these approximation methods are relatively easy to solve, their results can be substantially suboptimal. In this study, we discover that the proximal operator of the tubal rank can be explicitly solved. Then, without relaxing the tubal rank, an efficient proximal gradient algorithm is proposed to directly solve the tensor recovery problem. We establish the convergence properties of the proposed method and present an adaptive strategy for parameter selection. Experimental results on synthetic and real-world data demonstrate the superior performance of the proposed method over several widely used state-of-the-art methods in the literature. [ABSTRACT FROM AUTHOR]

Abstract Tensor decomposition is widely used to exploit the internal correlation in multi-way data analysis and process for communications and radar systems. As one of the main tensor decomposition methods, CANDECOMP/PARAFAC decomposition has advantages of uniqueness and interpretation properties which are significant in practical applications. However, traditional decomposition method is sensitive to both predefined rank and noise that results in inaccurate tensor decomposition. In this paper, we propose a improved algorithm called the Element-wise Average Alternating Direction Method of Multipliers by minimizing the sum of all factors’ trace norm and the noise variance. Our algorithm could overcome the dependence on predefined rank in traditional decomposition algorithms and alleviate the impact of noise. Moreover, this algorithm can be transferred to solve the problem of tensor completion conveniently. The simulation results show that our proposed algorithm could decompose the noisy tensor to the factors with above 90% similarity in various SNR and also interpolate the incomplete tensor with higher similar coefficient and lower relative reconstruction error when the missing rate is less than 0.5.

In recent years, the problem of incomplete data has been behind the appearance of several completion methods and algorithms. The truncated nuclear norm has been known as a powerful low-rank approach both for the matrix and the tensor cases. However, the low-rank approaches are unable to characterize some additional information exhibited in data such as the smoothness or feature-preserving properties. In this work, a tensor completion model based on the convex truncated nuclear norm and the nonconvex-sparse total variation is introduced. Notably, we develop an alternating minimization algorithm that combines the accelerating proximal gradient for the convex step and a pro jection operator for the nonconvex step to solve the optimization problem. Experiments and comparative results show that our algorithm has a significant impact on the completion process. [ABSTRACT FROM AUTHOR]

The ubiquitous 5G-enable industrial Internet of Things interconnects a great number of intelligent sensors and actuators. 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.

Tensor decomposition is widely used to exploit the internal correlation in multi-way data analysis and process for communications and radar systems. As one of the main tensor decomposition methods, CANDECOMP/PARAFAC decomposition has advantages of uniqueness and interpretation properties which are significant in practical applications. However, traditional decomposition method is sensitive to both predefined rank and noise that results in inaccurate tensor decomposition. In this paper, we propose a improved algorithm called the Element-wise Average Alternating Direction Method of Multipliers by minimizing the sum of all factors' trace norm and the noise variance. Our algorithm could overcome the dependence on predefined rank in traditional decomposition algorithms and alleviate the impact of noise. Moreover, this algorithm can be transferred to solve the problem of tensor completion conveniently. The simulation results show that our proposed algorithm could decompose the noisy tensor to the factors with above 90% similarity in various SNR and also interpolate the incomplete tensor with higher similar coefficient and lower relative reconstruction error when the missing rate is less than 0.5. [ABSTRACT FROM AUTHOR]

Aiming at recovering an unknown tensor (i.e., multi-way array) corrupted by both sparse outliers and dense noises, robust tensor decomposition (RTD) serves as a powerful pre-processing tool for subsequent tasks like classification and target detection in many computer vision and machine learning applications. Recently, tubal nuclear norm (TNN) based optimization is proposed with superior performance as compared with other tensorial nuclear norms for tensor recovery. However, one major limitation is its orientation sensitivity due to low-rankness strictly defined along tubal orientation and it cannot simultaneously model spectral low-rankness in multiple orientations. To this end, we introduce two new tensor norms called OITNN-O and OITNN-L to exploit multi-orientational spectral low-rankness for an arbitrary K-way (K ≥ 3) tensors. We further formulate two RTD models via the proposed norms and develop two algorithms as the solutions. Theoretically, we establish non-asymptotic error bounds which can predict the scaling behavior of the estimation error. Experiments on real-world datasets demonstrate the superiority and effectiveness of the proposed norms. [ABSTRACT FROM AUTHOR]

Tensor compressed sensing (TCS) from binary measurements aims to recover a tensor with low-rankness from the binary quantization of its degraded linear measurements, which inherits the advantages of low equipment cost and high sampling efficiency of coarse quantization. Recently, numerous studies have demonstrated that incorporating the properties of low-rankness and local smoothness, which are commonly observed in practical tensor data, can significantly enhance conventional tensor recovery tasks. This motivates us to utilize the low-rank and smooth properties in solving TCS problems from binary measurements. Equipped with the recently proposed tensor correlated total variation (TCTV) that simultaneously regularizes low-rankness and smoothness, we present a regularized model for this problem in this paper. We rigorously prove the recovery guarantee for the proposed model and provide an explicit selection rule for the regularization parameter. Moreover, the theoretical results demonstrate the robustness of the proposed method against both bit flips and pre-quantization noises. An algorithm based on alternating direction multiplier method (ADMM) is presented for solving the proposed model, with its global convergence been proven. Finally, a series of experiments demonstrate the superior performance of our method compared to many other competing approaches. • A method for tensor recovery from binary measurements which codes both low-rankness and smoothness was proposed. • We prove a recovery guarantee for the method and provide an explicit selection rule for the regularization parameter. • The method is robust to bit flips and pre-quantization noises, two common noises in one-bit quantization. • The proposed model can be solved using the alternating direction multiplier method with global convergence. [ABSTRACT FROM AUTHOR]

Low-rank tensor recovery faces challenges in accurately defining the low-rankness of a tensor. Most existing definitions typically focus on one domain alone — either the original or frequency domain. Additionally, certain definitions often exhibit limitations in their sensitivity to orientation variation. To overcome these challenges, we define a novel tensor rank, the Orientation Invariant Hybrid Rank (OIHR). This rank fuses rank information across all orientations in both frequency and original domains. Employing its convex approximation, the Orientation Invariant Hybrid Nuclear Norm (OIHNN), we propose a general tensor recovery model. We further explore the statistical performance of the estimator based on this model, establishing a deterministic upper bound on the estimation error under generic noise. Furthermore, non-asymptotic upper bounds under Gaussian noise are separately derived for two specific cases: tensor compressive sensing and tensor completion. Finally, we propose the algorithm to solve the model. Extensive experiments on both synthetic and real data are conducted to validate the statistical guarantees and verify the effectiveness of our algorithm. • Our tensor rank fuses low-rankness on all orientations in frequency–original domain. • Tensor recovery algorithm is obtained and outperforms existing convex methods. • Statistical Guarantees are established for estimated error. [ABSTRACT FROM AUTHOR]

Statistical process control techniques have been widely used for online process monitoring and diagnosis of streaming data in various applications, including manufacturing, healthcare, and environmental engineering. In some applications, the sensing system that collects online data can only provide partial information from the process due to resource constraints. In such cases, an adaptive sampling strategy is needed to decide where to collect data while maximizing the change detection capability. This article proposes an adaptive sampling strategy for online monitoring and diagnosis with partially observed data. The proposed methodology integrates two novel ideas (i) the recursive projection of the high-dimensional streaming data onto a low-dimensional subspace to capture the spatio-temporal structure of the data while performing missing data imputation; and (ii) the development of an adaptive sampling scheme, balancing exploration and exploitation, to decide where to collect data at each acquisition time. Through simulations and two case studies, the proposed framework's performance is evaluated and compared with benchmark methods. [ABSTRACT FROM AUTHOR]

Abstract Higher-order tensors can represent scores in a rating system, frames in a video, and images of the same subject. In practice, the measurements are often highly quantized due to the sampling strategies or the quality of devices. Existing works on tensor recovery have focused on data losses and random noises. Only a few works consider tensor recovery from quantized measurements but are restricted to binary measurements. This paper, for the first time, addresses the problem of tensor recovery from multi-level quantized measurements by leveraging the low CANDECOMP/PARAFAC (CP) rank property. We study the recovery of both general low-rank tensors and tensors that have tensor singular value decomposition (TSVD) by solving nonconvex optimization problems. We provide the theoretical upper bounds of the recovery error, which diminish to zero when the sizes of dimensions increase to infinity. We further characterize the fundamental limit of any recovery algorithm and show that our recovery error is nearly order-wise optimal. A tensor-based alternating proximal gradient descent algorithm with a convergence guarantee and a TSVD-based projected gradient descent algorithm are proposed to solve the nonconvex problems. Our recovery methods can also handle data losses and do not necessarily need the information of the quantization rule. The methods are validated on synthetic data, image datasets, and music recommender datasets.

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 structures, many tensor recovery methods have been developed to apply various regularization techniques together with the operator-splitting type of algorithms. Due to the unprecedented growth of data, it becomes increasingly desirable to use streamlined algorithms to achieve real-time computation, such as stochastic optimization algorithms that have recently emerged as an efficient family of methods in machine learning. In this work, we propose a novel algorithmic framework based on the Kaczmarz algorithm for tensor recovery. We provide thorough convergence analysis and its applications from the vector case to the tensor one. Numerical results on a variety of tensor recovery applications, including sparse signal recovery, low-rank tensor recovery, image inpainting, and deconvolution, illustrate the enormous potential of the proposed methods. [ABSTRACT FROM AUTHOR]

In tensor recovery problem, larger singular values often correspond to the primary information of the image, such as contours, sharp edges and smooth areas. The minimum–maximum concave penalty (MCP) function has been effective in preserving larger singular values and achieving good tensor recovery results. However, in the process of solving this problem, singular values may vary during iterations, and the fixed parameters of the MCP function may not sufficiently preserve all the larger singular values, which may hinder the attainment of optimal results in tensor recovery. To overcome this challenge, we propose the Bivariate Equivalent Minimax-Concave Penalty (BEMCP) theorem, which allows the parameters to adapt to the changes in singular values and more comprehensively preserve the larger singular values. For low-rank tensor completion and tensor robust principal component analysis problems, we propose BEMCP-based models. Finally, experiments with various real-world data demonstrate that the proposed methods outperform state-of-the-art methods. • Overcome the challenge that the MCP function cannot vary with iterations. • A new equivalent theorem, known as the BEMCP theorem, is proposed. • Tensor forms of the TBEMCP theorem and the BEWTGN theorem are proposed. • For LRTC and TRPCA problem, two new BEMCP-based models are proposed respectively. • Numerical experiments demonstrate the superiority of the proposed method. [ABSTRACT FROM AUTHOR]

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 factor tensor has two low dimensions. We call such a decomposition the triple decomposition, and the corresponding rank the triple rank. The triple rank of a third order tensor is not greater than the middle value of the Tucker rank. The number of parameters in the bilevel form of standard triple decomposition is less than the number of parameters of Tucker decomposition in substantial cases. The theoretical discovery is confirmed numerically. Numerical tests show that third order tensor data from practical applications such as internet traffic and image are of low triple ranks. A tensor recovery method based on low rank triple decomposition is proposed. Its convergence and convergence rate are established. Numerical experiments confirm the efficiency of this method. [ABSTRACT FROM AUTHOR]

Tensor robust principal component analysis via tensor nuclear norm (TNN) minimization has been recently proposed to recover the low-rank tensor corrupted with sparse noise/outliers. TNN is demonstrated to be a convex surrogate of rank. However, it tends to over-penalize large singular values and thus usually results in biased solutions. To handle this issue, we propose a new definition of tensor logarithmic norm (TLN) as the nonconvex surrogate of rank, which can decrease the penalization on larger singular values and increase that on smaller ones simultaneously to preserve the low-rank structure of a tensor. Then, the strategy of tensor factorization is combined into the minimization of TLN to improve computational performance. To handle impulsive scenarios, we propose a nonconvex $\ell _{p}$ -ball projection scheme with $0 < p < 1$ instead of the conventional convex scheme with $p=1$ , which enhances the robustness against outliers. By incorporating the TLN minimization and the $\ell _{p}$ -ball projection, we finally propose two low-rank recovery algorithms, whose resulting optimization problems are efficiently solved by the alternating direction method of multipliers (ADMM) with convergence guarantees. The proposed algorithms are applied to the synthetic data recovery and image and video restorations in real-world. Experimental results demonstrate the superior performance of the proposed methods over several state-of-the-art algorithms in terms of tensor recovery accuracy and computational efficiency. [ABSTRACT FROM AUTHOR]

Low-rank tensor completion and recovery have received considerable attention in the recent literature. The existing algorithms, however, are prone to suffer a failure when the multiway data are simultaneously contaminated by arbitrary outliers and missing values. In this paper, we study the unsupervised tensor learning problem, in which a low-rank tensor is recovered from an incomplete and grossly corrupted multidimensional array. We introduce a unified framework for this problem by using a simple equation to replace the linear projection operator constraint, and further reformulate it as two convex optimization problems through different approximations of the tensor rank. Two globally convergent algorithms, derived from the alternating direction augmented Lagrangian (ADAL) and linearized proximal ADAL methods, respectively, are proposed for solving these problems. Experimental results on synthetic and real-world data validate the effectiveness and superiority of our methods. [ABSTRACT FROM AUTHOR]

Cauchy noise removal in multidimension visual data is a fundamental problem in image processing and data science communities. Recently, this problem can be effectively solved by resorting to an appropriate tensor recovery approach and has attracted increasing attention. However, it is not extensively reported whether tensor recovery approaches with multiple priors can achieve even better denosing performance. To address this issue, we propose a novel tensor recovery approach in this paper based on the maximum a posteriori estimation (MAP). Our approach is a nonconvex optimization model that includes a composite regularization term and a data fidelity term. The composite regularization term incorporates nonlocal self-similarity in the spatial and gradient domains of multidimensional visual data, preserving both local image details and non-local structures. The data fidelity term is obtained by computing the negative log-likelihood of Cauchy noise distribution. To solve this proposed tensor recovery model, we develop an effective denoising algorithm with a global convergence guarantee via the well-known nonconvex alternating direction method of multipliers (ADMM). Finally, we conduct extensive numerical experiments to demonstrate that our proposed recovery approach can achieve superior denoising performance, both quantitatively and qualitatively, compared to existing common methods. [ABSTRACT FROM AUTHOR]

Product and content personalization is now ubiquitous in e-commerce. There are typically not enough available transactional data for this task. As such, companies today seek to use a variety of information on the interactions between a product and a customer to drive personalization decisions. We formalize this problem as one of recovering a large-scale matrix with side information in the form of additional matrices of conforming dimension. Viewing the matrix we seek to recover and the side information we have as slices of a tensor, we consider the problem of slicerecovery, which is to recover specific slices of "simple" tensors from noisy observations of the entire tensor. We propose a definition of simplicity that on the one hand elegantly generalizes a standard generative model for our motivating problem and on the other hand subsumes low-rank tensors for a variety of existing definitions of tensor rank. We provide an efficient algorithm for slice recovery that is practical for massive data sets and provides a significant performance improvement over state-of-the-art incumbent approaches to tensor recovery. Furthermore, we establish near-optimal recovery guarantees that, in an important regime, represent an order improvement over the best available results for this problem. Experiments on data from a music streaming service demonstrate the performance and scalability of our algorithm. The e-companion is available at https://doi.org/10.1287/mnsc.2018.3092. [ABSTRACT FROM AUTHOR]

The growing popularity of tensor singular value decomposition (T-SVD) in tensor recovery problems and the N -tubal rank can be applied to higher order tensors. However, this method faces new challenges as it does not fully use the rank prior information of tensors and lacks non-convex relaxation methods. Thus, this paper applies a new non-convex function, called Minimax Logarithmic Concave Penalty (MLCP), based on the N -tubal rank method. Two MLCP models based on N -tubal rank are introduced for solving low-rank tensor completion (LRTC) and tensor robust principal component analysis (TRPCA) problems, along with corresponding solving algorithms. The effectiveness and superiority of these models are demonstrated through experiments on real datasets. [ABSTRACT FROM AUTHOR]

The dominant multipath components for massive multiple-input multiple-output systems can be described using geometry-based channel models with R-dimensional (R-D) parameters. These parameters are crucial for channel correlation acquisition, which is a prerequisite for many technical challenges. In this paper, we consider higher-order singular value decomposition based R-D channel modeling parameter estimation from incomplete measurements. Incomplete higher-order orthogonality iteration algorithm can be utilized to solve the problem, which simultaneously achieves tensor recovery and tensor decomposition. After obtaining the signal or noise subspace, the parameters of interest can be estimated by using subspace methods. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the CP-nuclear value of a completely positive (CP) tensor and study its properties. A semidefinite relaxation algorithm is proposed for solving the minimal CP-nuclear-value tensor recovery. If a partial tensor is CP-recoverable, the algorithm can give a CP tensor recovery with the minimal CP-nuclear value, as well as a CP-nuclear decomposition of the recovered CP tensor. If it is not CP-recoverable, the algorithm can always give a certificate for that, when it is regular. Some numerical experiments are also presented. [ABSTRACT FROM AUTHOR]

Detecting anomalous traffic is a critical task for advanced Internet management. Many anomaly detection algorithms have been proposed recently. However, constrained by their matrix-based traffic data model, existing algorithms often suffer from low accuracy in anomaly detection. To fully utilize the multi-dimensional information hidden in the traffic data, this paper takes the initiative to investigate the potential and methodologies of performing tensor factorization for more accurate Internet anomaly detection. More specifically, we model the traffic data as a three-way tensor and formulate the anomaly detection problem as a robust tensor recovery problem with the constraints on the rank of the tensor and the cardinality of the anomaly set. These constraints, however, make the problem extremely hard to solve. Rather than resorting to the convex relaxation at the cost of low detection performance, we propose TensorDet to solve the problem directly and efficiently. To improve the anomaly detection accuracy and tensor factorization speed, TensorDet exploits the factorization structure with two novel techniques, sequential tensor truncation and two-phase anomaly detection. We have conducted extensive experiments using Internet traffic trace data Abilene and GÈANT. Compared with the state of art algorithms for tensor recovery and matrix-based anomaly detection, TensorDet can achieve significantly lower false positive rate and higher true positive rate. Particularly, benefiting from our well designed algorithm to reduce the computation cost of tensor factorization, the tensor factorization process in TensorDet is 5 (Abilene) and 13 (GÈANT) times faster than that of the traditional Tucker decomposition solution. [ABSTRACT FROM PUBLISHER]

Since matrix representation of video data could damage its initial structure, this paper proposed a patch-based denoising method based on low-rank tensor recovery-Frist, it constructed a three order tensor through clustering similar patches in the preprocessing video sequences. Then according to low-rank property of video tensor and sparsity of noise artifacts，the proposed approach used the augmented Lagrange multipliers (ALM) to reconstruct the low-rank and sparse sensors, which could completely separate noise from the video tensor. This paper developed a tensor model to preserve the spatial structure of the video modality，thus it could remove the noise artifacts from complex video better. Simulation experiments show that this algorithm has the stronger ability of video denoising comparing with traditional methods. [ABSTRACT FROM AUTHOR]

Abstract: The problem of recovering data in multi-way arrays (i.e., tensors) arises in many fields such as image processing and computer vision, etc. In this paper, we present a novel method based on multi-linear n-rank and norm optimization for recovering a low-n-rank tensor with an unknown fraction of its elements being arbitrarily corrupted. In the new method, the n-rank and norm of the each mode of the given tensor are combined by weighted parameters as the objective function. In order to avoid relaxing the observed tensor into penalty terms, which may cause less accuracy problem, the minimization problem along each mode is accomplished by applying the augmented Lagrange multiplier method. In experiments, we test the influence of parameters on the results of the proposed method, and then compare with one state-of-the-art method on both simulated data and real data. Numerical results show that the method can reliably solve a wide range of problems at a speed at least several times faster than the state-of-the-art method while the results of the method are comparable to the previous method in terms of accuracy. [Copyright &y& Elsevier]

This work proposes a systematic model reduction approach based on rank adaptive tensor recovery for partial differential equation (PDE) models with high-dimensional random parameters. Since the standard outputs of interest of these models are discrete solutions on given physical grids which are high-dimensional, we use kernel principal component analysis to construct stochastic collocation approximations in reduced dimensional spaces of the outputs. To address the issue of high-dimensional random inputs, we develop a new efficient rank adaptive tensor recovery approach to compute the collocation coefficients. Novel efficient initialization strategies for non-convex optimization problems involved in tensor recovery are also developed in this work. We present a general mathematical framework of our overall model reduction approach, analyze its stability, and demonstrate its efficiency with numerical experiments. [ABSTRACT FROM AUTHOR]

• A TNN-based least square estimator is proposed for robust tensor decomposition. We only assume the underlying tensor to satisfy the l ∞ -norm boundedness condition, which is less strict than the tensor incoherence conditions. • On the statistical side, both deterministic and non-asymptotic upper bounds on the estimation error are established in Theorems 1 and 2 , respectively. The non-asymptotic upper bounds are then proved to be minimax near-optimal by Theorem 3. Experiments on synthetic dataset verify that the proposed upper bounds can predict the scaling behavior of the estimation error. • On the computational side, two algorithms, i.e., an ADMM-based algorithm (Algorithm 1) and an FW-based algorithm (Algorithm 1), are proposed with convergence guarantees (Theorems 4 and 5). The latter gets rid of the proximity operator of TNN, and has significantly cheaper one-iteration computational cost. Experiments on real dataset validate the effectiveness and the efficiency of the proposed algorithms. Aiming at recovering a signal tensor from its mixture with outliers and noises, robust tensor decomposition (RTD) arises frequently in many real-world applications. Recently, the low-tubal-rank model has shown more powerful performances than traditional tensor low-rank models in several tensor recovery tasks. Assuming the underlying tensor to be low-tubal-rank and the outliers sparse, this paper first proposes a penalized least squares estimator for RTD. Specifically, we adopt the tubal nuclear norm (TNN) and a sparsity inducing norm to regularize the underlying tensor and the outliers, respectively. Then, from a statistical standpoint, non-asymptotic upper bounds on the estimation error are established and proved to be near-optimal in a minimax sense. Further, two algorithms, namely, an ADMM-based algorithm and a Frank-Wolfe (FW) based algorithm are proposed to efficiently solve the proposed estimator from a computational standpoint. The sharpness of the proposed upper bound is verified on synthetic datasets. The superiority and efficiency of the proposed algorithms is demonstrated in experiments on real datasets. [ABSTRACT FROM AUTHOR]