In this paper, we study the problem of low-rank tensor learning, where only a few of training samples are observed and the underlying tensor has a low-rank structure. The existing methods are based on the sum of nuclear norms of unfolding matrices of a tensor, which may be suboptimal. In order to explore the low-rankness of the underlying tensor effectively, we propose a nonconvex model based on transformed tensor nuclear norm for low-rank tensor learning. Specifically, a family of nonconvex functions are employed onto the singular values of all frontal slices of a tensor in the transformed domain to characterize the low-rankness of the underlying tensor. An error bound between the stationary point of the nonconvex model and the underlying tensor is established under restricted strong convexity on the loss function (such as least squares loss and logistic regression) and suitable regularity conditions on the nonconvex penalty function. By reformulating the nonconvex function into the difference of two convex functions, a proximal majorization-minimization (PMM) algorithm is designed to solve the resulting model. Then the global convergence and convergence rate of PMM are established under very mild conditions. Numerical experiments are conducted on tensor completion and binary classification to demonstrate the effectiveness of the proposed method over other state-of-the-art methods.

Hyspectral images (HSIs) are inevitably degraded by a mixture of different types of noise, such as Gaussian noise, impulse noise, stripes, and deadlines. Traditional methods exploit the global low-rankness and local smoothness by using two separate regularization terms for HSI denoising, which may be highly affected by the trade-off parameters between the two regularizers. In this paper, we propose to utilize the three directional tensor correlated total variation (3DTCTV) to explore the global low-rankness and local smoothness of HSIs simultaneously. More specifically, the three directional transformed tensor nuclear norm is employed to the gradient domain of the underlying HSIs along the spatial and spectral dimensions. And the tensor ℓ1norm is used to characterize the sparsity of the noisy tensor. Moreover, a nonconvex approach is proposed to characterize the low-rankness of the underlying HSIs and enhance the sparsity of noisy tensor further for HSI denoising, where a family of nonconvex functions are employed onto the singular values of the three directional gradient tensors along spatial-spectral dimensions in the transformed domain and each entry of the sparse tensor, respectively. An alternating direction method of multipliers is developed to solve the resulting models and its convergence for the nonconvex model is established under some conditions. Extensive numerical experiments on simulated and real HSI datasets demonstrate that the proposed methods outperform various state-of-the-art methods both visually and quantitatively.

Robust tensor completion, which aims to recover a tensor from partial observations corrupted by Gaussian noise and sparse noise simultaneously, has a wide range of applications in visual data recovery. The existing approaches make use of convex or nonconvex relaxation based on transformed tensor nuclear norm, which may be challenged since only the global low-rankness of the underlying tensor data is utilized. In order to explore the global and local patterns simultaneously, in this paper, we propose a nonconvex model for robust tensor completion by combining the dictionary learning and nonconvex regularization. For the sake of exploring the global low-rankness, a family of nonconvex functions are employed onto the singular values of all frontal slices of the underlying tensor in the transformed domain. The dictionary learning is utilized to elucidate the local patterns of the underlying tensor data. Moreover, a family of nonconvex functions are used onto each entry of the sparse noise, which can obtain sparser solutions compared with tensor$\ell _{1}$norm. A proximal alternating linearized minimization algorithm is designed to solve the proposed model, whose convergence is established under very mild conditions. Extensive numerical experiments on multispectral images, video, and magnetic resonance imaging datasets show that the proposed model outperforms other state-of-the-art approaches.

Abstract. The transform-based tensor nuclear norm (TNN) methods have shown good recovery results for tensor completion. However, the TNN methods are based on the single-tube transforms in which transforms are applied to each tube independently. The performance of the single-tube transform-based TNN methods is not good for recovery of missing tubes in multidimensional images (e.g., all the observations are missing in a pixel location of multispectral images). The main aim of this paper is to address this issue by proposing and developing a learnable group-tube transform-based TNN (GTNN) method that can effectively explore the correlation of neighboring tubes by leveraging a learnable group-tube transform. The proposed learnable group-tube transform is a separable three-dimensional transform that consists of a one-dimensional spectral/temporal transform (i.e., single-tube transform) and a two-dimensional spatial transform. Such group-tube transform can effectively explore the correlation of neighboring tubes. Based on the elaborately designed low-rank metric GTNN, we suggest a low-rank tensor completion model. To solve this highly nonconvex model, we design an efficient multiblock proximal alternating minimization algorithm and establish the convergence guarantee. A variety of numerical experiments on real-world multidimensional imaging data including traffic speed data, color images, videos, and multispectral images collectively manifest that the GTNN method outperforms some state-of-the-art TNN methods especially when the observations along tubes are missing.

In this article, we study the robust tensor completion problem in three-dimensional image data, where only partial entries are available and the observed tensor is corrupted by Gaussian noise and sparse noise simultaneously. Compared with the existing tensor nuclear norm minimization for the low-rank component, we propose to use the transformed tensor nuclear norm to explore the global low-rankness of the underlying tensor. Moreover, the plug-and-play (PnP) deep prior denoiser is incorporated to preserve the local details of multi-dimensional images. Besides, the tensor$\ell _{1}$norm is utilized to characterize the sparseness of the sparse noise. A symmetric Gauss-Seidel based alternating direction method of multipliers is designed to solve the resulting model under the PnP framework with deep prior denoiser. Extensive numerical experiments on hyperspectral and multispectral images, videos, color images, and magnetic resonance image datasets are conducted to demonstrate the superior performance of the proposed model in comparison with several state-of-the-art models.

Poisson observations for videos are important models in video processing and computer vision. In this paper, we study the third-order tensor completion problem with Poisson observations. The main aim is to recover a tensor based on a small number of its Poisson observation entries. A existing matrix-based method may be applied to this problem via the matricized version of the tensor. However, this method does not leverage on the global low-rankness of a tensor and may be substantially suboptimal. Our approach is to consider the maximum likelihood estimate of the Poisson distribution, and utilize the Kullback-Leibler divergence for the data-fitting term to measure the observations and the underlying tensor. Moreover, we propose to employ a transformed tensor nuclear norm ball constraint and a bounded constraint of each entry, where the transformed tensor nuclear norm is used to get a lower transformed multi-rank tensor with suitable unitary transformation matrices. We show that the upper bound of the error of the estimator of the proposed model is less than that of the existing matrix-based method. Also an information theoretic lower error bound is established. An alternating direction method of multipliers is developed to solve the resulting convex optimization model. Extensive numerical experiments on synthetic data and real-world datasets are presented to demonstrate the effectiveness of our proposed model compared with existing tensor completion methods.

In this paper, we study the sparse nonnegative tensor factorization and completion problem from partial and noisy observations for third-order tensors. Because of sparsity and nonnegativity, the underlying tensor is decomposed into the tensor-tensor product of one sparse nonnegative tensor and one nonnegative tensor. We propose to minimize the sum of the maximum likelihood estimation for the observations with nonnegativity constraints and the tensor$\ell _{0}$norm for the sparse factor. We show that the error bounds of the estimator of the proposed model can be established under general noise observations. The detailed error bounds under specific noise distributions including additive Gaussian noise, additive Laplace noise, and Poisson observations can be derived. Moreover, the minimax lower bounds are shown to be matched with the established upper bounds up to a logarithmic factor of the sizes of the underlying tensor. These theoretical results for tensors are better than those obtained for matrices, and this illustrates the advantage of the use of nonnegative sparse tensor models for completion and denoising. Numerical experiments are provided to validate the superiority of the proposed tensor-based method compared with the matrix-based approach.

In this paper, we study the problem of low-rank tensor recovery from limited sampling with noisy observations for third-order tensors. A tensor nuclear norm method based on a convex relaxation of the tubal rank of a tensor has been used and studied for tensor completion. In this paper, we propose to incorporate a corrected term in the tensor nuclear norm method for tensor completion. Theoretically, we provide a nonasymptotic error bound of the corrected tensor nuclear norm model for low-rank tensor completion. Moreover, we develop and establish the convergence of a symmetric Gauss--Seidel based multiblock alternating direction method of multipliers to solve the proposed correction model. Extensive numerical examples on both synthetic and real-world data are presented to validate the superiority of the proposed model over several state-of-the-art methods.

Low-rank tensor completion plays an important role in many applications such as image processing, computer vision, and machine learning. A widely used convex relaxation of this problem is to minimize the nuclear norm of the square deal matrix generated by reshaping a tensor. However, this approach can be substantially suboptimal. In order to seek a highly accurate solution, in this paper, we propose to use a family of nonconvex functions onto the singular values of the square deal matrix of the tensor to approximate the rank of the tensor. A proximal linearized minimization (PLM) algorithm is proposed to solve the resulting model. Furthermore, based on the Kurdyka-Łojasiewicz property, we show that the sequence generated by the PLM algorithm globally converges to a critical point of the objective function. Extensive numerical experiments including synthetic data, video data, and the extended Yale Face Database B show the effectiveness of the proposed model compared with several existing state-of-the-art models.

Orthogonal nonnegative matrix factorization plays an important role for data clustering and machine learning. In this paper, we propose a new optimization model for orthogonal nonnegative matrix factorization based on the structural properties of orthogonal nonnegative matrix. The new model can be solved by a novel convex relaxation technique which can be employed quite efficiently. Numerical examples in document clustering, image segmentation and hyperspectral unmixing are used to test the performance of the proposed model. The performance of our method is better than the other testing methods in terms of clustering accuracy.

In this paper, we propose and study a nonconvex data fitting term and a total variation regularization term for image restoration with impulse noise removal. The proposed model is different from existing image restoration models where the data fitting term is based on the $\ell_1$- or $\ell_2$- norm, and the regularization term is based on the total variation, $\ell_1$-norm, or some nonconvex functions. Theoretically, we analyze the properties of minimizers of the proposed objective function with the nonconvex data fitting term and the total variation regularization term. We show that minimizers can preserve piecewise constant regions or match with the data points perfectly. This property is particularly useful for impulse noise removal. The proposed image restoration model can be solved by the proximal linearized minimization algorithm, and the global convergence of the iterative algorithm can also be established according to Kurdyka--Łojasiewicz property. The performance of the proposed model is tested for image restoration with salt-and-pepper impulse noise or random-valued impulse noise. We demonstrate that the restored images by the proposed Nonconvex-TV model are better (in terms of PSNR and visual quality) than those by the other existing data fitting plus regularization models, including $\ell_1$ plus total variation (L1TV) and $\ell_1$ plus nonconvex (L1Nonconvex) methods.

In this paper, we study the tensor completion problem on recovery of the multilinear data under limited sampling. A popular convex relaxation of this problem is to minimize the nuclear norm of the more square matrix produced by matricizing a tensor. However, it may fail to produce a highly accurate solution under low sample ratio. In order to get a recovery with high accuracy, we propose an adaptive correction approach for tensor completion. First, a corrected model for matrix completion with bound constraint is proposed and its error bound is established. Then, we extend it to tensor completion with bound constraint and propose a corrected model for tensor completion. The adaptive correction approach consists of solving a series of corrected models with an initial estimator where the initial estimator used for the next step is computed from the value of the current solution. Moreover, the error bound of the corrected model for tensor completion is also established. A convergent 3-block alternating direction method of multipliers (ADMM) is applied to solve the dual problem of the corrected model. Numerical experiments on both random data and real world data validate the efficiency of our proposed correction approach.