Low-rank Tensor Completion Research Articles

Learnable Transform-Assisted Tensor Decomposition for Spatio-Irregular Multidimensional Data Recovery

Tensor decompositions have been successfully applied to multidimensional data recovery. However, classical tensor decompositions are not suitable for emerging spatio-irregular multidimensional data (i.e., spatio-irregular tensor), whose spatial domain is non-rectangular, e.g., spatial transcriptomics data from bioinformatics and semantic units from computer vision. By using preprocessing (e.g., zero-padding or element-wise 0-1 weighting), the spatio-irregular tensor can be converted to a spatio-regular tensor and then classical tensor decompositions can be applied, but this strategy inevitably introduces bias information, leading to artifacts. How to design a tensor-based method suitable for emerging spatio-irregular tensors is an imperative challenge. To address this challenge, we propose a learnable transform-assisted tensor singular value decomposition (LTA-TSVD) for spatio-irregular tensor recovery, which allows us to leverage the intrinsic structure behind the spatio-irregular tensor. Specifically, we design a learnable transform to project the original spatio-irregular tensor into its latent spatio-regular tensor, and then the latent low-rank structure is captured by classical TSVD on the resulting regular tensor. Empowered by LTA-TSVD, we develop spatio-irregular low-rank tensor completion (SIR-LRTC) and spatio-irregular tensor robust principal component analysis (SIR-TRPCA) models for the spatio-irregular tensor imputation and denoising respectively, and we design corresponding solving algorithms with theoretical convergence. Extensive experiments including the spatial transcriptomics data imputation and hyperspectral image denoising show SIR-LRTC and SIR-TRPCA are superior performance to competing approaches and benefit downstream applications.

Adaptive sampling with tensor leverage scores for exact low-rank third-order tensor completion

Tensor completion aims at estimating the missing entries from the incomplete observation. Under the tensor singular value decomposition framework, the exact recovery of a low-tubal-rank third-order tensor could be achieved via convex optimization with high probability if the tensor satisfies the tensor incoherence condition. In this work, we show that, when the random selection of entries is made adaptive to a distribution which is dependent on the coherence structure of the tensor, any low-tubal-rank tensor of the size n×n×n with tubal-rank r can be exactly recovered with high probability from as few as O(rn2log2⁡(n)) randomly chosen entries. In practice, tensor leverage scores are not known a priori, and we design a two-phase adaptive sampling strategy to obtain the leverage scores. Numerical experiments on synthetic and real-world third-order tensor data sets are used to validate our theoretical results and illustrate that the tensor recovery performance of the proposed two-phase adaptive sampling scheme is better than that of the other state-of-the-art methods.

Laplacian-enhanced non-synchronous measurements method

To address the challenges of physical geometric constraints of microphone arrays and the lack of a prior information about the operating frequencies of target signals, we propose a novel method for broadband multiple-sound-source localization: a non-synchronous measurements method for broadband multiple-sound-source localization based on Laplacian-enhanced low-rank tensor completion. Our method thoroughly analyzes the tensor data structure of broadband signals and introduces a multiplier-optimized alternating direction method. Through extensive simulation studies, our approach has achieved remarkable results. We validate the outstanding performance of the Laplacian-enhanced algorithm, demonstrating its ability to generate highly accurate sound maps, capturing five distinct speech signal sources. Compared to other methods, our approach provides a more comprehensive global view. This technological advancement brings substantial hope for the precise localization of multiple broadband sound sources in critical applications, offering significant potential for future applications.

Low-rank tensor completion based on tensor train rank with partially overlapped sub-blocks and total variation

Recently, the low-rank tensor completion method based on tensor train (TT) rank has achieved promising performance. Ket augmentation (KA) is commonly used in TT rank-based methods to improve the performance by converting low-dimensional tensors to higher-dimensional tensors. However, block artifacts are caused since KA also destroys the original structure and image continuity of original low-dimensional tensors. To tackle this issue, a low-rank tensor completion method based on TT rank with tensor augmentation by partially overlapped sub-blocks (TAPOS) and total variation (TV) is proposed in this paper. The proposed TAPOS preserves the image continuity of the original tensor and enhances the low-rankness of the generated higher-dimensional tensors, and a weighted de-augmentation method is used to assign different weights to the elements of sub-blocks and further reduce the block artifacts. To further alleviate the block artifacts and improve reconstruction accuracy, TV is introduced in the TAPOS-based model to add the piecewise smooth prior. The parallel matrix decomposition method is introduced to estimate the TT rank to reduce the computational cost. Numerical experiments show that the proposed method outperforms the existing state-of-the-art methods.

Low-rank tensor completion via nonlocal self-similarity regularization and orthogonal transformed tensor Schatten-p norm

Low-rank tensor completion via nonlocal self-similarity regularization and orthogonal transformed tensor Schatten-p norm

A novel ship trajectory reconstruction approach based on low-rank tensor completion

The Automatic Identification System (AIS) enhances maritime safety and environmental monitoring by providing crucial ship trajectory data. However, this data is often compromised by missing or abnormal values due to signal blockage, transmission errors, and equipment failures, jeopardizing safety and the accuracy of environmental analyses. Addressing these challenges, we propose a novel Customized Tensor-Based Maritime Trajectory Reconstruction (CTMTR) framework that leverages the self-similarity of ship trajectories to reformulate trajectory reconstruction as a matrix rank minimization issue. The CTMTR framework consists of three steps: data preprocessing, trajectory matrix construction, and trajectory reconstruction. We conducted simulation experiments using a dataset comprising vessel trajectories from the east coast of Dover in the United States in 2022. To validate its effectiveness, the CTMTR is compared with four advanced methods (LRMC, LRTC-TNN, TRPCA, and TNN-DCT) under diverse missing and anomalous scenarios. The results substantiate that the performance of CTMTR outperforms other approaches, especially in anomalous cases. Our method outperforms existing methods by an order of magnitude under random missing combined with anomaly scenarios. The CTMTR framework thus has the potential to advance maritime trajectory reconstruction methodologies, providing a solid foundation for future innovations in maritime data analysis and navigation safety technologies.

Efficient enhancement of low-rank tensor completion via thin QR decomposition.

Low-rank tensor completion (LRTC), which aims to complete missing entries from tensors with partially observed terms by utilizing the low-rank structure of tensors, has been widely used in various real-world issues. The core tensor nuclear norm minimization (CTNM) method based on Tucker decomposition is one of common LRTC methods. However, the CTNM methods based on Tucker decomposition often have a large computing cost due to the fact that the general factor matrix solving technique involves multiple singular value decompositions (SVDs) in each loop. To address this problem, this article enhances the method and proposes an effective CTNM method based on thin QR decomposition (CTNM-QR) with lower computing complexity. The proposed method extends the CTNM by introducing tensor versions of the auxiliary variables instead of matrices, while using the thin QR decomposition to solve the factor matrix rather than the SVD, which can save the computational complexity and improve the tensor completion accuracy. In addition, the CTNM-QR method's convergence and complexity are analyzed further. Numerous experiments in synthetic data, real color images, and brain MRI data at different missing rates demonstrate that the proposed method not only outperforms in terms of completion accuracy and visualization, but also conducts more efficiently than most state-of-the-art LRTC methods.

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.

A generalizable framework for low-rank tensor completion with numerical priors

Low-Rank Tensor Completion, a method which exploits the inherent structure of tensors, has been studied extensively as an effective approach to tensor completion. Whilst such methods attained great success, none have systematically considered exploiting the numerical priors of tensor elements. Ignoring numerical priors causes loss of important information regarding the data, and therefore prevents the algorithms from reaching optimal accuracy. Despite the existence of some individual works which consider ad hoc numerical priors for specific tasks, no generalizable frameworks for incorporating numerical priors have appeared. We present the Generalized CP Decomposition Tensor Completion (GCDTC) framework, the first generalizable framework for low-rank tensor completion that takes numerical priors of the data into account. We test GCDTC by further proposing the Smooth Poisson Tensor Completion (SPTC) algorithm, an instantiation of the GCDTC framework, whose performance exceeds current state-of-the-arts by considerable margins in the task of non-negative tensor completion, exemplifying GCDTC’s effectiveness. Our code is open-source.

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.

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.

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.

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.

Low-Rank Tensor Completion Pansharpening Based on Haze Correction

Low-Rank Tensor Completion Pansharpening Based on Haze Correction

A neural tensor decomposition model for high-order sparse data recovery

Tensor decomposition has attracted wide attention in the low-rank tensor completion (LRTC) problem because of its marvelous recovering ability to missing entries. However, previous LRTC methods are generally based on linear and shallow models, which are prone to overfitting when the data is sparse, resulting in significantly degraded performance. Meanwhile, the models are highly sensitive and suffer from the difficult rank selection problem. To address these issues, we propose an effective and novel tensor-ring (TR) decomposition method based on the convolutional computation (ConvTR), which can be regarded as a natural extension of deep learning models for the LRTC problem. Specifically, ConvTR employs a multi-layer convolutional neural network (CNN) to model the complex interactions between TR factors. Each element in the index vector of the observation tensor can be embedded as a corresponding tensor slice in the factor tensor decomposed by the TR model. These individual slice matrices are then concatenated to get a wider matrix used for extracting the nonlinear features by feeding them into a 2D convolutional layer. A fully-connected layer is utilized to aggregate the final convoluted features to a scalar value, which is the desired missing entry indexed by the original index vector exactly. Extensive experiments on various common datasets verified the effectiveness of the proposed method and demonstrated its superior to the traditional TR-based completion methods and other state-of-the-art network-based methods.

Structural-Missing Tensor Completion for Robust DOA Estimation with Sensor Failure

Array sensor failure poses a serious challenge to robust direction-of-arrival (DOA) estimation in complicated environments. Although existing matrix completion methods can successfully recover the damaged signals of an impaired sensor array, they cannot preserve the multi-way signal characteristics as the dimension of arrays expands. In this paper, we propose a structural-missing tensor completion algorithm for robust DOA estimation with uniform rectangular array (URA), which exhibits a high robustness to non-ideal sensor failure conditions. Specifically, the signals received at the impaired URA are represented as a three-dimensional incomplete tensor, which contains whole fibers or slices of missing elements. Due to this structural-missing pattern, the conventional low-rank tensor completion becomes ineffective. To resolve this issue, a spatio-temporal dimension augmentation method is developed to transform the structural-missing tensor signal into a six-dimensional Hankel tensor with dispersed missing elements. The augmented Hankel tensor can then be completed with a low-rank regularization by solving a Hankel tensor nuclear norm minimization problem. As such, the inverse Hankelization on the completed Hankel tensor recovers the tensor signal of an unimpaired URA. Accordingly, a completed covariance tensor can be derived and decomposed for robust DOA estimation. Simulation results verify the effectiveness of the proposed algorithm.

Joint tensor completion anchor selection and coordinate transformation for high precision cooperative positioning in device-to-device systems

The synergistic amalgamation of device-to-device (D2D) communication between mobile terminals (MT) and ultra-dense networks in 5G systems enables cooperative positioning. In this paper, we propose a joint tensor completion, anchor selection, and coordinate transformation (JTCASCT) for high precision cooperative positioning in D2D systems. The positioning is achieved by mitigating the challenging issue of recovering the missing values from the corrupted distance between nodes and determining the global positions of unknown stationary nodes. Initially, a hankelized distance tensor structure called Euclidean distance tensor (EDT) is constructed from incomplete EDM. The missing values are recovered as tensor completion by exploiting its latent low rank structure. The optimization problem is solved using simple low rank tensor completion (SiLRTC) and high accuracy low rank tensor completion (HaLRTC) algorithms. The relative position of nodes is determined after applying multidimensionality scaling (MDS). To reduce the data dimension for coordinate transformation and to select the optimal combination of anchor nodes, we propose a low complex Kruskal–Wallis (KW) test. Finally, the global position of unknown nodes is computed using Helmert transformation (HT). Numerical evaluations demonstrate that the cooperative positioning accuracy is significantly improved with the proposed JTCASCT approach compared to the state-of-the-art techniques.

NT-DPTC: A non-negative temporal dimension preserved tensor completion model for missing traffic data imputation

Missing traffic data imputation is an important step in the intelligent transportation systems. Low rank approximation is an important method for the missing traffic data imputation, especially the low rank matrix/tensor completion. However, the low-rank matrix completion has a huge time cost and the low-rank tensor completion cannot fully extract the information and maintain non-negativity. Therefore, we propose a novel non-negative temporal dimension preserved tensor completion (NT-DPTC) model with ideas: a) proposing a novel dimension preserved (DP) tensor decomposition method, which decomposes a tensor into three latent factor tensors to fully extract the intrinsic feature, b) using a Sigmoid mapper for releasing non-negative constraint from the training process and increasing flexibility, c) exploiting temporal constrains and AdamW schemes for obtaining a higher accuracy and faster convergence. Experiments on four industrial application scenarios highlight the superiority of our proposed model when compared with the existing state-of-the-art models.

Tensor recovery based on rank adaptive and non-convex methods

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.

Toward urban traffic scenarios and more: a spatio-temporal analysis empowered low-rank tensor completion method for data imputation

Existing traffic monitoring approaches cannot completely cover all road segments in real-time, leading to massive amounts of missing traffic data, which limits the implementation of intelligent transportation systems. Most existing methods lack deep mining of the unique spatiotemporal characteristics of traffic flows, resulting in difficulty in application to urban traffic with complex topologies and variable states. In this paper, we propose a novel Spatio-Temporal constrained Low-Rank Tensor Completion (ST-LRTC) method, which adopts a manifold embedding approach to depict the local geometric structure of spatiotemporal domains. Specifically, under the low-rank assumption, the method introduces temporal constraints based on the continuity and periodicity of traffic flow and a spatial constraint matrix reflecting the traffic flow transmission mechanism. We embed low-dimensional spatiotemporal constraint matrices into the low-rank tensor completion solving process to fully utilize the global features and local spatiotemporal characteristics of the traffic tensor. Experiments were performed using traffic data from Xi’an, China, and the results indicated that ST-LRTC outperformed state-of-the-art methods under various missing rates and patterns. Thorough experiments have demonstrated that the incorporation of spatiotemporal analysis can enhance the adaptability of the tensor completion model to complex urban scenarios, which guarantees better monitoring, diagnosis, and optimization of urban traffic states.