Tensor Completion Research Articles

Tensor completion via multi-directional partial tensor nuclear norm with total variation regularization

Tensor completion via multi-directional partial tensor nuclear norm with total variation regularization

A Flexible and Robust Tensor Completion Approach for Traffic Data Recovery With Low-Rankness

A Flexible and Robust Tensor Completion Approach for Traffic Data Recovery With Low-Rankness

Nonnegative tensor completion for dynamic counterfactual prediction on COVID-19 pandemic

Nonnegative tensor completion for dynamic counterfactual prediction on COVID-19 pandemic

Quaternion tensor completion with sparseness for color video recovery

A novel low-rank completion algorithm based on the quaternion tensor is proposed in this paper. This approach uses the TQt-rank of quaternion tensor to maintain the structure of RGB channels throughout the entire process. In more detail, the pixels in each frame are encoded on three imaginary parts of a quaternion as an element in a quaternion matrix. Each quaternion matrix is then stacked into a quaternion tensor. A logarithmic function and truncated nuclear norm are employed to characterize the rank of the quaternion tensor in order to promote the low rankness of the tensor. Moreover, by introducing a newly defined quaternion tensor discrete cosine transform-based (QTDCT) regularization to the low-rank approximation framework, the optimized recovery results can be obtained in the local details of color videos. In particular, the sparsity of the quaternion tensor is reasonably characterized by l1 norm in the QDCT domain. This strategy is optimized via the two-step alternating direction method of multipliers (ADMM) framework with convergence analysis. Numerical experimental results for recovering color videos show the obvious advantage of the proposed method over other potential competing approaches.

Distributed neural tensor completion for network monitoring data recovery

Network monitoring data is usually incomplete, accurate and fast recovery of missing data is of great significance for practical applications. The tensor-based nonlinear methods have attracted recent attentions with their capability of capturing complex interactions among data for more accurate recovery. However, the training process of existing methods is often time-consuming due to massive data and unreasonable network resource allocation. Thus motivated, we propose a distributed neural tensor completion method, named D-NORM, which simultaneously optimizes both recovery accuracy and time. Specifically, D-NORM adopts two schemes to solve the resulting optimization problem. First, we design a parameter-efficient multi-layer architecture with convolutional neural network to learn nonlinear correlations among data. Second, we reformulate the initial model as an equivalent set function optimization problem under a matroid base constraint. After constructing an approximate supermodular function to substitute the objective set function with provable upper bound, we propose an approximation algorithm based on the two-stage search procedure with theoretical performance guarantee to rationally allocate computing resources and efficiently recover missing data. Extensive experiments conducted on real-world datasets validate the superiority of D-NORM in both efficiency and effectiveness.

Tensor completion via joint reweighted tensor Q-nuclear norm for visual data recovery

Recently, the transform-based tensor nuclear norm methods have achieved encouraging results for low-rank tensor completion (LRTC) under the tensor singular value decomposition (t-SVD) framework. Among them, the tensor Q-nuclear norm, which uses a data-dependent matrix Q as transform, is more flexible than that of using fixed transform when handling different types of data. However, it only describes the spectral correlations and ignores the spatial dimensions’ information. Besides, it disregards the necessity for unbalanced singular value penalty, which may lead to the loss of primary information and inadequate sparsity of singular values in the recovery results. To overcome the above defects, this paper presents a new definition of tensor rank, called tensor joint Q-rank, via the proposed tensor decomposition, i.e., the mode-k Q-T-SVD. In addition, we adopt a joint reweighted tensor Q-nuclear norm (JRTQN) as its non-convex relaxation, with a novel reweighted strategy and data-dependent transforms Qk(k=1,2,3) along each mode. What is more, based on the low-rank assumption, we provide a method to choose Qk by maximizing the variance of singular value distribution. Then, we propose a JRTQN-TC model, solved via the alternating direction multipliers method and the theoretical convergence is guaranteed. Extensive experiments carried out on color image and video recovery, multispectral image inpainting, face image completion and CT and MRI image restoration demonstrate the highly competitive performance of the proposed method quantitatively and visually, compared with the related methods.

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.

A Novel Tensor Ring Sparsity Measurement for Image Completion.

As a promising data analysis technique, sparse modeling has gained widespread traction in the field of image processing, particularly for image recovery. The matrix rank, served as a measure of data sparsity, quantifies the sparsity within the Kronecker basis representation of a given piece of data in the matrix format. Nevertheless, in practical scenarios, much of the data are intrinsically multi-dimensional, and thus, using a matrix format for data representation will inevitably yield sub-optimal outcomes. Tensor decomposition (TD), as a high-order generalization of matrix decomposition, has been widely used to analyze multi-dimensional data. In a direct generalization to the matrix rank, low-rank tensor modeling has been developed for multi-dimensional data analysis and achieved great success. Despite its efficacy, the connection between TD rank and the sparsity of the tensor data is not direct. In this work, we introduce a novel tensor ring sparsity measurement (TRSM) for measuring the sparsity of the tensor. This metric relies on the tensor ring (TR) Kronecker basis representation of the tensor, providing a unified interpretation akin to matrix sparsity measurements, wherein the Kronecker basis serves as the foundational representation component. Moreover, TRSM can be efficiently computed by the product of the ranks of the mode-2 unfolded TR-cores. To enhance the practical performance of TRSM, the folded-concave penalty of the minimax concave penalty is introduced as a nonconvex relaxation. Lastly, we extend the TRSM to the tensor completion problem and use the alternating direction method of the multipliers scheme to solve it. Experiments on image and video data completion demonstrate the effectiveness of the proposed method.

Tensor recovery using the tensor nuclear norm based on nonconvex and nonlinear transformations

Recently, the tensor completion problem has witnessed significant advancements with the introduction of the tensor nuclear norm based on self-supervised nonlinear transformations (S2NTNN). However, the tensor nuclear norm serves as a convex substitute for the tubal rank of tensors, often resulting in suboptimal solutions to the original problem. In this work, we propose a novel method, called Nonconvex Nonlinear transformation-based Tensor Nuclear Norm (N2TNN), which builds upon nonlinear transformations and introduces a generalized nonconvex function to better exploit the prior information of underlying tensors. Additionally, we utilize regularization techniques, namely spatial total variation (TV) and spatial spectral total variation (SSTV), to enhance both the details and edge information. Extensive numerical experiments unequivocally validate the superior performance of our proposed method when compared to other state-of-the-art techniques in terms of both numerical accuracy and visual quality.

A Novel Truncated Normal Tensor Completion Method for Multi-Source Fusion Data

Completing traffic data is a basic requirement for intelligent transportation systems. However, completing spatiotemporal traffic data poses a significant challenge, especially for high-dimensional data with complex missing mechanisms. Various completion methods targeting different missing mechanisms have showcased the superiority of tensor learning by effectively characterizing intricate spatiotemporal correlations. In this study, a novel tensor completion framework, known as the multi-source tensor completion method for data fusion, is proposed. This framework incorporates passenger transfer relationships between buses and subways into subway data completion, enhancing the data completion accuracy. Moreover, by combining bus transfer passenger flow data with other data dimensions, such as the different road sections, time intervals, and days, an innovative 4D low-rank tensor completion data framework was obtained. In addition, to boost the completion accuracy, a truncated l2,p norm optimization model was derived. This model enhances the non-convex performance of the objective function throughout the tensor completion process. The results highlight the superiority of the proposed completion method, leveraging fused subway/bus data over other completion methods that rely solely on subway data.

Spectrum Situation Awareness for Space-Air-Ground Integrated Networks Based on Tensor Computing.

The spectrum situation awareness problem in space-air-ground integrated networks (SAGINs) is studied from a tensor-computing perspective. Tensor and tensor computing, including tensor decomposition, tensor completion and tensor eigenvalues, can satisfy the application requirements of SAGINs. Tensors can effectively handle multidimensional heterogeneous big data generated by SAGINs. Tensor computing is used to process the big data, with tensor decomposition being used for dimensionality reduction to reduce storage space, and tensor completion utilized for numeric supplementation to overcome the missing data problem. Notably, tensor eigenvalues are used to indicate the intrinsic correlations within the big data. A tensor data model is designed for space-air-ground integrated networks from multiple dimensions. Based on the multidimensional tensor data model, a novel tensor-computing-based spectrum situation awareness scheme is proposed. Two tensor eigenvalue calculation algorithms are studied to generate tensor eigenvalues. The distribution characteristics of tensor eigenvalues are used to design spectrum sensing schemes with hypothesis tests. The main advantage of this algorithm based on tensor eigenvalue distributions is that the statistics of spectrum situation awareness can be completely characterized by tensor eigenvalues. The feasibility of spectrum situation awareness based on tensor eigenvalues is evaluated by simulation results. The new application paradigm of tensor eigenvalue provides a novel direction for practical applications of tensor theory.

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.

Sensor-independent LAI/FPAR CDR: reconstructing a global sensor-independent climate data record of MODIS and VIIRS LAI/FPAR from 2000 to 2022

Abstract. Leaf area index (LAI) and fraction of photosynthetically active radiation (FPAR) are critical biophysical parameters for the characterization of terrestrial ecosystems. Long-term global LAI/FPAR products, such as the moderate resolution imaging spectroradiometer (MODIS) and the Visible Infrared Imaging Radiometer Suite (VIIRS), provide the fundamental dataset for accessing vegetation dynamics and studying climate change. However, existing global LAI/FPAR products suffer from several limitations, including spatial–temporal inconsistencies and accuracy issues. Considering these limitations, this study develops a sensor-independent (SI) LAI/FPAR climate data record (CDR) based on Terra-MODIS/Aqua-MODIS/VIIRS LAI/FPAR standard products. The SI LAI/FPAR CDR covers the period from 2000 to 2022, at spatial resolutions of 500 m/5 km/0.05∘, 8 d/bimonthly temporal frequencies and available in sinusoidal and WGS1984 projections. The methodology includes (i) comprehensive analyses of sensor-specific quality assessment variables to select high-quality retrievals, (ii) application of the spatial–temporal tensor (ST-tensor) completion model to extrapolate LAI and FPAR beyond areas with high-quality retrievals, (iii) generation of SI LAI/FPAR CDR in various projections and various spatial and temporal resolutions, and (iv) evaluation of the CDR by direct comparisons with ground data and indirectly through reproducing results of LAI/FPAR trends documented in the literature. This paper provides a comprehensive analysis of each step involved in the generation of the SI LAI/FPAR CDR, as well as evaluation of the ST-tensor completion model. Comparisons of SI LAI (FPAR) CDR with ground truth data suggest an RMSE of 0.84 LAI (0.15 FPAR) units with R2 of 0.72 (0.79), which outperform the standard Terra/Aqua/VIIRS LAI (FPAR) products. The SI LAI/FPAR CDR is characterized by a low time series stability (TSS) value, suggesting a more stable and less noisy dataset than sensor-dependent counterparts. Furthermore, the mean absolute error (MAE) of the CDR is also lower, suggesting that SI LAI/FPAR CDR is comparable in accuracy to high-quality retrievals. LAI/FPAR trend analyses based on the SI LAI/FPAR CDR agree with previous studies, which indirectly provides enhanced capabilities to utilize this CDR for studying vegetation dynamics and climate change. Overall, the integration of multiple satellite data sources and the use of advanced gap filling modeling techniques improve the accuracy of the SI LAI/FPAR CDR, ensuring the reliability of long-term vegetation studies, global carbon cycle modeling, and land policy development for informed decision-making and sustainable environmental management. The SI LAI/FPAR CDR is open access and available under a Creative Commons Attribution 4.0 License at https://doi.org/10.5281/zenodo.8076540 (Pu et al., 2023a).

Coarse to Fine Two-Stage Approach to Robust Tensor Completion of Visual Data.

Tensor completion is the problem of estimating the missing values of high-order data from partially observed entries. Data corruption due to prevailing outliers poses major challenges to traditional tensor completion algorithms, which catalyzed the development of robust algorithms that alleviate the effect of outliers. However, existing robust methods largely presume that the corruption is sparse, which may not hold in practice. In this article, we develop a two-stage robust tensor completion approach to deal with tensor completion of visual data with a large amount of gross corruption. A novel coarse-to-fine framework is proposed which uses a global coarse completion result to guide a local patch refinement process. To efficiently mitigate the effect of a large number of outliers on tensor recovery, we develop a new M-estimator-based robust tensor ring recovery method which can adaptively identify the outliers and alleviate their negative effect in the optimization. The experimental results demonstrate the superior performance of the proposed approach over state-of-the-art robust algorithms for tensor completion.

Nonconvex Robust High-Order Tensor Completion Using Randomized Low-Rank Approximation.

Within the tensor singular value decomposition (T-SVD) framework, existing robust low-rank tensor completion approaches have made great achievements in various areas of science and engineering. Nevertheless, these methods involve the T-SVD based low-rank approximation, which suffers from high computational costs when dealing with large-scale tensor data. Moreover, most of them are only applicable to third-order tensors. Against these issues, in this article, two efficient low-rank tensor approximation approaches fusing random projection techniques are first devised under the order-d ( d ≥ 3 ) T-SVD framework. Theoretical results on error bounds for the proposed randomized algorithms are provided. On this basis, we then further investigate the robust high-order tensor completion problem, in which a double nonconvex model along with its corresponding fast optimization algorithms with convergence guarantees are developed. Experimental results on large-scale synthetic and real tensor data illustrate that the proposed method outperforms other state-of-the-art approaches in terms of both computational efficiency and estimated precision.

Noisy Tensor Completion via Low-Rank Tensor Ring.

Tensor completion is a fundamental tool for incomplete data analysis, where the goal is to predict missing entries from partial observations. However, existing methods often make the explicit or implicit assumption that the observed entries are noise-free to provide a theoretical guarantee of exact recovery of missing entries, which is quite restrictive in practice. To remedy such drawback, this article proposes a novel noisy tensor completion model, which complements the incompetence of existing works in handling the degeneration of high-order and noisy observations. Specifically, the tensor ring nuclear norm (TRNN) and least-squares estimator are adopted to regularize the underlying tensor and the observed entries, respectively. In addition, a nonasymptotic upper bound of estimation error is provided to depict the statistical performance of the proposed estimator. Two efficient algorithms are developed to solve the optimization problem with convergence guarantee, one of which is specially tailored to handle large-scale tensors by replacing the minimization of TRNN of the original tensor equivalently with that of a much smaller one in a heterogeneous tensor decomposition framework. Experimental results on both synthetic and real-world data demonstrate the effectiveness and efficiency of the proposed model in recovering noisy incomplete tensor data compared with state-of-the-art tensor completion models.

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

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

Tensor completion using high-order spatial delay embedding for IoT multi-attribute data reconstruction

Tensor completion using high-order spatial delay embedding for IoT multi-attribute data reconstruction

Differentially Private Federated Tensor Completion for Cloud–Edge Collaborative AIoT Data Prediction

Differentially Private Federated Tensor Completion for Cloud–Edge Collaborative AIoT Data Prediction

Sparse Bayesian Tensor Completion for Data Recovery in Intelligent IoT Systems

Sparse Bayesian Tensor Completion for Data Recovery in Intelligent IoT Systems