Tensor Recovery Problem Research Articles

Tensor recovery based on Bivariate Equivalent Minimax-Concave Penalty

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.

Research on Infrared Dim and Small Target Detection Algorithm Based on Low-Rank Tensor Recovery

In order to rapidly and accurately detect infrared small and dim targets in the infrared image of complex scene collected by virtual prototyping of space-based downward-looking multiband detection, an improved detection algorithm of infrared small and dim target is proposed in this paper. Firstly, the original infrared images are changed into a new infrared patch tensor mode through data reconstruction. Then, the infrared small and dim target detection problems are converted to low-rank tensor recovery problems based on tensor nuclear norm in accordance with patch tensor characteristics, and inverse variance weighted entropy is defined for self-adaptive adjustment of sparseness. Finally, the low-rank tensor recovery problem with noise is solved by alternating the direction method to obtain the sparse target image, and the final small target is worked out by a simple partitioning algorithm. The test results in various space-based downward-looking complex scenes show that such method can restrain complex background well by virtue of rapid arithmetic speed with high detection probability and low false alarm rate. It is a kind of infrared small and dim target detection method with good performance.

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.

Tensor Recovery Based on a Novel Non-Convex Function Minimax Logarithmic Concave Penalty Function.

Non-convex relaxation methods have been widely used in tensor recovery problems, compared with convex relaxation methods, and can achieve better recovery results. In this paper, a new non-convex function, Minimax Logarithmic Concave Penalty (MLCP) function, is proposed, and some of its intrinsic properties are analyzed, among which it is interesting to find that the Logarithmic function is an upper bound of the MLCP function. The proposed function is generalized to tensor cases, yielding tensor MLCP and weighted tensor Lγ -norm. Consider that its explicit solution cannot be obtained when applying it directly to the tensor recovery problem. Therefore, the corresponding equivalence theorems to solve the such problem are given, namely, tensor equivalent MLCP theorem and equivalent weighted tensor Lγ -norm theorem. In addition, we propose two EMLCP-based models for classic tensor recovery problems, namely low-rank tensor completion (LRTC) and tensor robust principal component analysis (TRPCA), and design proximal alternate linearization minimization (PALM) algorithms to solve them individually. Furthermore, based on the Kurdyka-Åasiwicz property, it is proved that the solution sequence of the proposed algorithm has a finite length and converges to the critical point globally. Finally, extensive experiments show that the proposed algorithm achieves good results, and it is confirmed that the MLCP function is indeed better than the Logarithmic function in the minimization problem, which is consistent with the analysis of theoretical properties.

Learning-Based High-Frame-Rate SAR Imaging

As high-frame-rate synthetic aperture radar (SAR) has the ability to form continuous SAR images and dynamically monitor the ground areas of interest, it has attracted more and more attention nowadays. In practical application, the enormous data in high-frame-rate SAR system to obtain the multi-frame images brings big challenges to its transmission, storage, and processing. In order to solve this problem, there are many recent papers on formulating the high-frame-rate SAR imaging problem into a low-rank tensor recovery problem, and correspondingly, the sampling amount of the high-frame-rate SAR data can be largely reduced. However, existing algorithms to solve the low-rank tensor recovery problem in high-frame-rate SAR application still suffer from large computational cost. Under the above inspiration, this paper proposes a deep neural network architecture for high-frame-rate SAR imaging, i.e., Tensor Alternating Direction Method of Multiplier Network (TADMM-Net), which is more computational efficient in imaging procedure. Specifically, we formulate the high-frame-rate SAR imaging processing into a low-tubal-rank tensor recovery problem. We solve the low-tubal-rank tensor recovery problem using a tensor Alternating Direction Method of Multiplier (ADMM) algorithm and then design a new deep neural network architecture by applying algorithm unfolding techniques to the underlying low-rank tensor recovery problem. The proposed TADMM-Net approach shifts the computational burden from the testing phase to the training phase, and the practical processing time can be extremely decreased compared with the existing algorithms for the low-rank tensor recovery problem in high-frame-rate SAR imaging application. It also offers various advantages over the existing high-frame-rate SAR imaging algorithms including higher performance in the case with low sampling amount, lower storage complexity and no requirement for hand-craft hyper-parameters adjustment. The methodology was tested on high-frame-rate SAR data. These tests show that the proposed architecture outperforms other state-of-the-art methods in high-frame-rate SAR imaging applications.

Tensor Spectral k-Support Norm Minimization for Detecting Infrared Dim and Small Target Against Urban Backgrounds

In the low-altitude urban background with heavy interference, especially in the face of corner interference with higher intensity than the target, infrared (IR) dim and small target is extremely lack of prior information (i.e., size, shape and contrast information). In such case, the existing detection methods usually suffer from high false alarm or even failure. To deal with this situation, we develop a novel spatial-temporal tensor model with tensor spectral <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> -support norm minimization (STTM-TSNM) for detecting IR dim and small target. Firstly, the spatial-temporal information of the original image sequence can be preserved completely by constructing the holistic STTM. Then, according to the spatial-temporal related prior knowledge of the target and background, the target detection task is customized as an optimization problem of low rank and sparse tensor recovery. To better preserve the internal structure and capture more global information, the tensor spectral <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> -support norm minimization is introduced as the regularization term of the constraint background. Finally, draw support from the framework of alternating direction method of multipliers (ADMM) algorithm, the precise separation of target and background is achieved. In addition, to promote the prosperity of sequential detection methods, we released to the scientific community a small IR target dataset containing six image sequences with urban background. The experimental results on six real IR sequences demonstrate that our method outputs the most outstanding detection performance compared with the latest sequential detection methods.

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.

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.

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.

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.

Video SAR Imaging Based on Low-Rank Tensor Recovery

Due to its ability of forming continuous images for a ground scene of interest, the video synthetic aperture radar (SAR) has been studied in recent years. However, as video SAR needs to reconstruct many frames, the data are of enormous amount and the imaging process is of large computational cost, which limits its applications. In this article, we exploit the redundancy property of multiframe video SAR data, which can be modeled as low-rank tensor, and formulate the video SAR imaging process as a low-rank tensor recovery problem, which is solved by an efficient alternating minimization method. We empirically compare the proposed method with several state-of-the-art video SAR imaging algorithms, including the fast back-projection (FBP) method and the compressed sensing (CS)-based method. Experiments on both simulated and real data show that the proposed low-rank tensor-based method requires significantly less amount of data samples while achieving similar or better imaging performance.

Robust Tensor Recovery with Fiber Outliers for Traffic Events

Event detection is gaining increasing attention in smart cities research. Large-scale mobility data serves as an important tool to uncover the dynamics of urban transportation systems, and more often than not the dataset is incomplete. In this article, we develop a method to detect extreme events in large traffic datasets, and to impute missing data during regular conditions. Specifically, we propose a robust tensor recovery problem to recover low-rank tensors under fiber-sparse corruptions with partial observations, and use it to identify events, and impute missing data under typical conditions. Our approach is scalable to large urban areas, taking full advantage of the spatio-temporal correlations in traffic patterns. We develop an efficient algorithm to solve the tensor recovery problem based on the alternating direction method of multipliers (ADMM) framework. Compared with existing l 1 norm regularized tensor decomposition methods, our algorithm can exactly recover the values of uncorrupted fibers of a low-rank tensor and find the positions of corrupted fibers under mild conditions. Numerical experiments illustrate that our algorithm can achieve exact recovery and outlier detection even with missing data rates as high as 40% under 5% gross corruption, depending on the tensor size and the Tucker rank of the low rank tensor. Finally, we apply our method on a real traffic dataset corresponding to downtown Nashville, TN and successfully detect the events like severe car crashes, construction lane closures, and other large events that cause significant traffic disruptions.

Tensor recovery from noisy and multi-level quantized measurements

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.

Multi-View Spectral Clustering Tailored Tensor Low-Rank Representation

This paper explores the problem of multi-view spectral clustering (MVSC) based on tensor low-rank modeling. Unlike the existing methods that all adopt an off-the-shelf tensor low-rank norm without considering the special characteristics of the tensor in MVSC, we design a novel structured tensor low-rank norm tailored to MVSC. Specifically, we explicitly impose a symmetric low-rank constraint and a structured sparse low-rank constraint on the frontal and horizontal slices of the tensor to characterize the intra-view and inter-view relationships, respectively. Moreover, the two constraints could be jointly optimized to achieve mutual refinement. On basis of the novel tensor low-rank norm, we formulate MVSC as a convex low-rank tensor recovery problem, which is then efficiently solved with an augmented Lagrange multiplier-based method iteratively. Extensive experimental results on seven commonly used benchmark datasets show that the proposed method outperforms state-of-the-art methods to a significant extent. Impressively, our method is able to produce perfect clustering. In addition, the parameters of our method can be easily tuned, and the proposed model is robust to different datasets, demonstrating its potential in practice. The code is available at <uri xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">https://github.com/jyh-learning/MVSC-TLRR</uri> .

Improved robust tensor principal component analysis for accelerating dynamic MR imaging reconstruction.

Dynamic magnetic resonance imaging (dMRI) strikes a balance between reconstruction speed and image accuracy in medical imaging field. In this paper, an improved robust tensor principal component analysis (RTPCA) method is proposed to reconstruct the dynamic magnetic resonance imaging (MRI) from highly under-sampled K-space data. The MR reconstruction problem is formulated as a high-order low-rank tenor plus sparse tensor recovery problem, which is solved by robust tensor principal component analysis (RTPCA) with a new tensor nuclear norm (TNN). To further exploit the low-rank structures in multi-way data, the core matrix nuclear norm, extracted from the diagonal elements of the core tensor under tensor singular value decomposition (t-SVD) framework, is also integrated into TNN for enforcing the low-rank structure in MRI datasets. The experimental results show that the proposed method outperforms state-of-the-art methods in terms of both MR image reconstruction accuracy and computational efficiency on 3D and 4D experiment datasets, especially for 4D MR image reconstruction. Graphical abstract The flowchart of the proposed method to reconstruct the dynamic magnetic resonance imaging (MRI) from highly under-sampled K-space data in the kth iteration. To further exploit the low-rank structures in multi-way data, the core matrix nuclear norm, extracted from the diagonal elements of the core tensor under tensor singular value decomposition (t-SVD) framework, is also integrated into tensor nuclear norm (TNN) for enforcing the low-rank structure in MRI datasets. In each iteration, the first step is to get low-rank tensor ℓk - 1 by using soft thresholding on the singular values of ℓk - 1 = χk - 1 - ξk - 1, and an improved tensor nuclear norm method is proposed to process the low-rank tensor ℓk - 1 firstly. Then, the shrinkage operator is applied to ξk - 1 = χk - 1 - ℓk - 1 for sparse part ξk - 1. The final reconstructed d-MRI χk is obtained by enforcing data consistency that the residual in K-space is subtracted by the sum of the reconstructed low-rank tensor and sparse tensor.

On Recoverability of Randomly Compressed Tensors With Low CP Rank

Our interest lies in the recoverability properties of compressed tensors under the \textit{canonical polyadic decomposition} (CPD) model. The considered problem is well-motivated in many applications, e.g., hyperspectral image and video compression. Prior work studied this problem under somewhat special assumptions---e.g., the latent factors of the tensor are sparse or drawn from absolutely continuous distributions. We offer an alternative result: We show that if the tensor is compressed by a subgaussian linear mapping, then the tensor is recoverable if the number of measurements is on the same order of magnitude as that of the model parameters---without strong assumptions on the latent factors. Our proof is based on deriving a \textit{restricted isometry property} (R.I.P.) under the CPD model via set covering techniques, and thus exhibits a flavor of classic compressive sensing. The new recoverability result enriches the understanding to the compressed CP tensor recovery problem; it offers theoretical guarantees for recovering tensors whose elements are not necessarily continuous or sparse.

Iterative p-shrinkage thresholding algorithm for low Tucker rank tensor recovery

Low-rank tensor recovery, as a higher order extension of low-rank matrix recovery, has generated a great deal of research interests in recent years, such as image inpainting, video inpainting, video decoding, scan completion, and so on. In this paper, we propose an easy-to-implement algorithm based on the framework of alternative direction method, named iterative p-shrinkage thresholding algorithm, for solving the low Tucker rank tensor recovery problem. The performance of the proposed algorithm is investigated on both synthetic and real data. Numerical results on simulation data demonstrate that our algorithm can successfully recover varieties of synthetic low Tucker rank tensors in different sampling ratios with better quality compared to the existing state-of-the-art tensor recovery algorithms. Experiments on real data, including colored image inpainting, MRI image inpainting and hyperspectral image inpainting, further illustrate the effectiveness of the proposed iterative p-shrinkage thresholding algorithm.

A Corrected Tensor Nuclear Norm Minimization Method for Noisy Low-Rank Tensor Completion

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 t...

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.

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.