Low-rank Tensor Recovery Research Articles

Cyclic tensor singular value decomposition with applications in low-rank high-order tensor recovery

The rapid advancements in emerging technologies have increased the demand for recovery tasks involving high-dimensional data with complex structures. Effectively utilizing tensor decomposition techniques to capture the low-rank structure of such data is crucial. Recently, the high-order t-SVD has demonstrated strong adaptability. However, this decomposition approach can only capture the low-rank correlation of two modes along other modes individually, while disregarding the structural correlation between different modes. In this paper, we propose a novel cyclic tensor singular value decomposition (CTSVD) method that effectively characterizes the low-rank structures of high-order tensors along all modes. Specifically, our method decomposes an order-N tensor into N factor tensors and one core tensor, connecting them using a defined mode-k tensor-tensor product (t-product). Building upon this, we establish the corresponding tensor rank and its convex relaxation. To address the issue of dimensional imbalance between adjacent modes in high-dimensional data, we propose and integrate a square reshaping strategy into the recovery models for tensor completion (TC) and tensor principal component analysis (TRPCA) tasks. Effective alternating direction method of multipliers (ADMM)-based algorithms are designed to these tasks. Extensive experiments on both synthetic and real data demonstrate that our methods outperform state-of-the-art approaches.

TIHN: Tensor Improved Huber Norm for low-rank tensor recovery

Tensor Robust Principal Component Analysis (TRPCA) has received much attention in many real-world applications which aims to recover a low-rank tensor corrupted by sparse noise. Most existing TRPCA methods usually regularize the low-rank component by minimizing its Tensor Nuclear Norm (TNN). However, the original TNN shrinks all of the singular values with the same penalty which generally leads to suboptimal results. The reason is that the large singular values should be less penalized since they usually correspond to salient information of the real-world tensor. In this work, we develop a Tensor Improved Huber Norm (TIHN) which considers the difference between the singular values of tensor. The TIHN can achieve better performance by recovering the large and significant singular values exactly. We also establish the Huber TRPCA (HTRPCA) method by utilizing the proposed TIHN. In addition, an efficient optimization algorithm based on the half-quadratic theory and Alternating Direction Method of Multipliers (ADMM) framework is designed to implement the HTRPCA. Finally, experiments on color image recovery and video recovery validate the effectiveness of the proposed method.

Low-Rank Tensor Completion Based on Self-Adaptive Learnable Transforms.

The tensor nuclear norm (TNN), defined as the sum of nuclear norms of frontal slices of the tensor in a frequency domain, has been found useful in solving low-rank tensor recovery problems. Existing TNN-based methods use either fixed or data-independent transformations, which may not be the optimal choices for the given tensors. As the consequence, these methods cannot exploit the potential low-rank structure of tensor data adaptively. In this article, we propose a framework called self-adaptive learnable transform (SALT) to learn a transformation matrix from the given tensor. Specifically, SALT aims to learn a lossless transformation that induces a lower average-rank tensor, where the Schatten- p quasi-norm is used as the rank proxy. Then, because SALT is less sensitive to the orientation, we generalize SALT to other dimensions of tensor (SALTS), namely, learning three self-adaptive transformation matrices simultaneously from given tensor. SALTS is able to adaptively exploit the potential low-rank structures in all directions. We provide a unified optimization framework based on alternating direction multiplier method for SALTS model and theoretically prove the weak convergence property of the proposed algorithm. Experimental results in hyperspectral image (HSI), color video, magnetic resonance imaging (MRI), and COIL-20 datasets show that SALTS is much more accurate in tensor completion than existing methods. The demo code can be found at https://faculty.uestc.edu.cn/gaobin/zh_CN/lwcg/153392/list/index.htm.

Enhanced Low-Rank Tensor Recovery Fusing Reweighted Tensor Correlated Total Variation Regularization for Image Denoising

Enhanced Low-Rank Tensor Recovery Fusing Reweighted Tensor Correlated Total Variation Regularization for Image Denoising

Robust low tubal rank tensor recovery using discrete empirical interpolation method with optimized slice/feature selection

Robust low tubal rank tensor recovery using discrete empirical interpolation method with optimized slice/feature selection

Tensor robust PCA with nonconvex and nonlocal regularization

Tensor robust principal component analysis (TRPCA) is a classical way for low-rank tensor recovery, which minimizes the convex surrogate of tensor rank by shrinking each tensor singular value equally. However, for real-world visual data, large singular values represent more significant information than small singular values. In this paper, we propose a nonconvex TRPCA (N-TRPCA) model based on the tensor adjustable logarithmic norm. Unlike TRPCA, our N-TRPCA can adaptively shrink small singular values more and shrink large singular values less. In addition, TRPCA assumes that the whole data tensor is of low rank. This assumption is hardly satisfied in practice for natural visual data, restricting the capability of TRPCA to recover the edges and texture details from noisy images and videos. To this end, we integrate nonlocal self-similarity into N-TRPCA, and further develop a nonconvex and nonlocal TRPCA (NN-TRPCA) model. Specifically, similar nonlocal patches are grouped as a tensor and then each group tensor is recovered by our N-TRPCA. Since the patches in one group are highly correlated, all group tensors have strong low-rank property, leading to an improvement of recovery performance. Experimental results demonstrate that the proposed NN-TRPCA outperforms existing TRPCA methods in visual data recovery. The demo code is available at https://github.com/qguo2010/NN-TRPCA.

Statistical guaranteed noisy tensor recovery by fusing low-rankness on all orientations in frequency–original domains

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.

Attention-Guided Low-Rank Tensor Completion.

Low-rank tensor completion (LRTC) aims to recover missing data of high-dimensional structures from a limited set of observed entries. Despite recent significant successes, the original structures of data tensors are still not effectively preserved in LRTC algorithms, yielding less accurate restoration results. Moreover, LRTC algorithms often incur high computational costs, which hinder their applicability. In this work, we propose an attention-guided low-rank tensor completion (AGTC) algorithm, which can faithfully restore the original structures of data tensors using deep unfolding attention-guided tensor factorization. First, we formulate the LRTC task as a robust factorization problem based on low-rank and sparse error assumptions. Low-rank tensor recovery is guided by an attention mechanism to better preserve the structures of the original data. We also develop implicit regularizers to compensate for modeling inaccuracies. Then, we solve the optimization problem by employing an iterative technique. Finally, we design a multistage deep network by unfolding the iterative algorithm, where each stage corresponds to an iteration of the algorithm; at each stage, the optimization variables and regularizers are updated by closed-form solutions and learned deep networks, respectively. Experimental results for high dynamic range imaging and hyperspectral image restoration show that the proposed algorithm outperforms state-of-the-art algorithms.

A Multidimensional Tensor Low Rank Method for Magnetic Resonance Image Denoising.

In this paper, we present the Magnetic Resonance Image (MRI) denoising method via nonlocal multidimensional low rank tensor transformation constraint (NLRT). We first design a nonlocal MRI denoising method by non-local low rank tensor recovery framework. Furthermore, a multidimensional low rank tensor constraint is used to obtain low-rank prior information combined with 3-dimensional structure feature of MRI image cubes. Our NLRT can achieve denoising by retaining more image detail information. The optimization and updating process of the model is solved via the alternating direction method of multipliers (ADMM) algorithm. Several state-of-the-art denoising methods are selected for comparative experiments. In order to reflect the performance of the denoising method, Rician noise with different levels is added to the experiment to analyze the results. The experimental results prove that our NLTR has more outstanding denoising ability and can obtain better MRI images.

Low-rank tensor recovery for 2-D DOA estimation using cylindrical nested conformal array

In recent years, nested conformal arrays have received considerable attention because they are capable of providing extended array aperture and increased degrees of freedom compared to uniform conformal arrays. In this article, we suggest a low-rank tensor recovery algorithm for two-dimensional (2-D) direction-of-arrival (DOA) estimation with cylindrical nested conformal array (CNCA). First of all, we derive the tensor form coarray output of the CNCA and use the coarray output to construct a third-order augmented tensor. Subsequently, we reconstruct a noise-free third-order augmented tensor by formulating a tensor nuclear norm (TNN) minimization problem. With the reconstructed tensor, we finally derive the closed-form expressions of 2-D DOA estimates. Compared with the existing approaches, the proposed approach achieves better performance by exploiting the multidimensional structure inherent in the coarray output signal. Numerical results illustrate the advantage of the proposed method over several existing techniques.

Guaranteed Tensor Recovery Fused Low-rankness and Smoothness.

Tensor recovery is a fundamental problem in tensor research field. It generally requires to explore intrinsic prior structures underlying tensor data, and formulate them as certain forms of regularization terms for guiding a sound estimate of the restored tensor. Recent researches have made significant progress by adopting two insightful tensor priors, i.e., global low-rankness (L) and local smoothness (S), which are always encoded as a sum of two separate regularizers into recovery models. However, unlike the primary theoretical developments on low-rank tensor recovery, these joint "L+S" models have no theoretical exact-recovery guarantees yet, making the methods lack reliability in real practice. To this crucial issue, in this work, we build a unique regularizer termed as tensor correlated total variation (t-CTV), which essentially encodes both L and S priors of a tensor simultaneously. Especially, by equipping t-CTV into the recovery models, we can rigorously prove the exact recovery guarantees for two typical tensor recovery tasks, i.e., tensor completion and tensor robust principal component analysis. To the best of our knowledge, this should be the first exact-recovery results among all related "L+S" methods for tensor recovery. We further propose ADMM algorithms with fine convergence to solve the proposed models. Significant recovery accuracy improvements are observed in extensive experiments. Typically, our method achieves a workable performance when the missing rate is extremely large, e.g., 99.5%, for the color image inpainting task, while all its peers totally fail in such a challenging case. Code is released at https://github.com/wanghailin97.

Modewise operators, the tensor restricted isometry property, and low-rank tensor recovery

Modewise operators, the tensor restricted isometry property, and low-rank tensor recovery

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.

Hyperspectral Image Denoising via Weighted Multidirectional Low-Rank Tensor Recovery.

Recently, low-rank tensor recovery methods based on subspace representation have received increased attention in the field of hyperspectral image (HSI) denoising. Unfortunately, those methods usually analyze the prior structural information within different dimensions indiscriminately, ignoring the differences between modes, leaving substantial room for improvement. In this article, we first consider the low-rank properties in the subspace and prove that the structure correlation across the nonlocal self-similarity mode is much stronger than in the spatial sparsity and spectral correlation modes. On that basis, we introduce a new multidirectional low-rank regularization, in which each mode is assigned a different weight to characterize its contribution to estimating the tensor rank. After that, integrating the proposed regularization with the subspace-based tensor recovery framework, an optimization model for HSI mixed noise removal is developed. The proposed model can be addressed efficiently via the alternating minimization algorithm. Extensive experiments implemented with synthetic and real data demonstrate that the proposed method significantly outperforms other state-of-the-art HSI denoising methods, which clearly indicates the effectiveness of the proposed approach in HSI denoising.

Low-Tubal-Rank tensor recovery with multilayer subspace prior learning

Currently, low-rank tensor recovery employing the subspace prior information is an emerging topic, which has attracted considerable attention. However, existing studies cannot flexibly and fully utilize the accessible subspace prior information, thereby leading to suboptimal restored performance. Aiming at addressing this issue, based on the tensor singular value decomposition (t-SVD), this article presents a novel strategy that integrates more than two layers of subspace knowledge about columns and rows of target tensor into one unified recovery framework. Specially, we first design a multilayer subspace prior learning scheme, and then apply it to two common low-rank tensor recovery problems, i.e., tensor completion and tensor robust component principal analysis. Crucially, we prove that our approach can achieve exact recovery of tensors under a significantly weaker incoherence assumption than the analogous conditions previously proposed. Furthermore, two efficient algorithms with convergence guarantees based on alternating direction method of multipliers (ADMM) are proposed to solve the corresponding models. The experimental results on synthetic and real tensor data show that the proposed algorithms outperform other state-of-the-art algorithms in terms of both qualitative and quantitative metrics.

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.

Low-rank tensor recovery via non-convex regularization, structured factorization and spatio-temporal characteristics

Recently, the convex low-rank 3rd-order tensor recovery has attracted considerable attention. However, there are some limitations to the convex relaxation approach, which may yield biased estimators. To overcome this disadvantage, we develop a novel non-convex tensor pseudo-norm to replace the weighted sum of the tensor nuclear norm as a tighter rank approximation. Then in tensor robust principle component analysis, we introduce the noise analysis to separate the spare foreground from the dynamic background more accurately. Furthermore, by introducing a spatio-temporal matrix, we can make better use of the inherent spatio-temporal characteristics of the low-rank static background and sparse foreground. Finally, we introduce an incoherent term to constrain the sparse foreground and the dynamic background to improve the separability. Some preliminary numerical examples of color image, video, and face image data sets are presented to illustrate the efficiency of our proposed methods.

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.

Double Auto-weighted Tensor Robust Principal Component Analysis.

Tensor Robust Principal Component Analysis (TRPCA), which aims to recover the low-rank and sparse components from their sum, has drawn intensive interest in recent years. Most existing TRPCA methods adopt the tensor nuclear norm (TNN) and the tensor ℓ1 norm as the regularization terms for the low-rank and sparse components, respectively. However, TNN treats each singular value of the low-rank tensor L equally and the tensor ℓ1 norm shrinks each entry of the sparse tensor S with the same strength. It has been shown that larger singular values generally correspond to prominent information of the data and should be less penalized. The same goes for large entries in S in terms of absolute values. In this paper, we propose a Double Auto-weighted TRPCA (DATRPCA) method. Instead of using predefined and manually set weights merely for the low-rank tensor as previous works, DATRPCA automatically and adaptively assigns smaller weights and applies lighter penalization to significant singular values of the low-rank tensor and large entries of the sparse tensor simultaneously. We have further developed an efficient algorithm to implement DATRPCA based on the Alternating Direction Method of Multipliers (ADMM) framework. In addition, we have also established the convergence analysis of the proposed algorithm. The results on both synthetic and real-world data demonstrate the effectiveness of DATRPCA for low-rank tensor recovery, color image recovery and background modelling.

Nonconvex Tensor Completion for 5-D Seismic Data Reconstruction

Multidimensional pre-stack seismic data reconstruction can be viewed as a low-rank tensor completion problem. Recently, the nuclear norm has been widely used as a convex surrogate of the tensor rank function for low-rank tensor recovery and has been successfully applied to 5D seismic data reconstruction. However, solving the nuclear norm-based relaxed convex problem typically leads to a suboptimal solution of the original rank minimization problem, often degrading the reconstruction performance. In this study, to seek solutions to the aforementioned problems, we established a non-convex logDet function as a smooth approximation for the tensor rank instead of the convex tensor nuclear norm and applied it to solve the 5D seismic data reconstruction problem. Thereafter, we propose solving the obtained non-convex relaxation problem using an alternating direction method of multipliers (ADMM) algorithm. Numerical experiments of our approach on synthetic 5D seismic data demonstrated remarkable reconstruction performance compared with the performances of HOSVD, nulear norm, and PMF methods in terms of visual examination and numerical test. We further illustrate the performance of the proposed method using a land data survey.