Core Tensor Research Articles

Coupled Heterogeneous Tucker Decomposition: A Feature Extraction Method for Multisource Fusion and Domain Adaptation Using Multisource Heterogeneous Remote Sensing Data

To excavate adequately the rich information contained in multisource remote sensing data, feature extraction as basic yet important research has two typical applications: one of which is to extract complementary information of multisource data to improve classification; and the other is to extract shared information across sources for domain adaptation. However, typical feature extraction methods require the input represented as vectors or homogeneous tensors and fail to process multisource data represented as heterogeneous tensors. Therefore, the coupled heterogeneous Tucker decomposition (C-HTD) containing two sub-methods, namely coupled factor matrix-based HTD (CFM-HTD) and coupled core tensor-based HTD (CCT-HTD), is proposed to establish a unified feature extraction framework for multisource fusion and domain adaptation. To handle multisource heterogeneous tensors, multiple Tucker models were constructed to extract features of different sources separately. To cope with the supervised and semi-supervised cases, the class-indicator factor matrix was built to enhance the separability of features using known labels and learned labels. To mine the complementarity of paired multisource samples, coupling constraint was imposed on multiple factor matrices to form CFM-HTD to extract multisource information jointly. To extract domain-adapted features, coupling constraint was imposed on multiple core tensors to form CCT-HTD to encourage data from different sources to have the same class centroid. In addition, to reduce the impact of interference samples on domain adaptation, an adaptive sample-weighting matrix was designed to autonomously remove outliers. Using multiresolution multiangle optical and MSTAR datasets, experimental results show that the C-HTD outperforms typical multisource fusion and domain adaptation methods.

Winter Wheat Yield Estimation Based on Optimal Weighted Vegetation Index and BHT-ARIMA Model

This study aims to use remote sensing (RS) time-series data to explore the intrinsic relationship between crop growth and yield formation at different fertility stages and construct a high-precision winter wheat yield estimation model applicable to short time-series RS data. Sentinel-2 images were acquired in this study at six key phenological stages (rejuvenation stage, rising stage, jointing stage, heading stage, filling stage, filling-maturity stage) of winter wheat growth, and various vegetation indexes (VIs) at different fertility stages were calculated. Based on the characteristics of yield data continuity, the RReliefF algorithm was introduced to filter the optimal vegetation index combinations suitable for the yield estimation of winter wheat for all fertility stages. The Absolutely Objective Improved Analytic Hierarchy Process (AOIAHP) was innovatively proposed to determine the proportional contribution of crop growth to yield formation in six different phenological stages. The selected VIs consisting of MTCI(RE2), EVI, REP, MTCI(RE1), RECI(RE1), NDVI(RE1), NDVI(RE3), NDVI(RE2), NDVI, and MSAVI were then fused with the weights of different fertility periods to obtain time-series weighted data. For the characteristics of short time length and a small number of sequences of RS time-series data in yield estimation, this study applied the multiplexed delayed embedding transformation (MDT) technique to realize the data augmentation of the original short time series. Tucker decomposition was performed on the block Hankel tensor (BHT) obtained after MDT enhancement, and the core tensor was extracted while preserving the intrinsic connection of the time-series data. Finally, the resulting multidimensional core tensor was trained with the Autoregressive Integrated Moving Average (ARIMA) model to obtain the BHT-ARIMA model for wheat yield estimation. Compared to the performance of the BHT-ARIMA model with unweighted time-series data as input, the weighted time-series input significantly improves yield estimation accuracy. The coefficients of determination (R2) were improved from 0.325 to 0.583. The root mean square error (RMSE) decreased from 492.990 to 323.637 kg/ha, the mean absolute error (MAE) dropped from 350.625 to 255.954, and the mean absolute percentage error (MAPE) decreased from 4.332% to 3.186%. Besides, BHT-ARMA and BHT-CNN models were also used to compare with BHT-ARIMA. The results indicated that the BHT-ARIMA model still had the best yield prediction accuracy. The proposed method of this study will provide fast and accurate guidance for crop yield estimation and will be of great value for the processing and application of time-series RS data.

Block term decomposition with distinct time granularities for temporal knowledge graph completion

To solve the incompleteness problem in temporal knowledge graphs (TKGs) and to discover the new knowledge, TKG completion remains an essential task always solved by graph embedding technology. Existing TKG completion methods encode time at only single granularity, which is insufficient in exploiting the rich information of distinct time granularities. Furthermore, most of them lack a comprehensive consideration of the characteristic of both time points and time periods, resulting in the inability to handle the two types of facts with different time forms, namely the discrete facts and continuous facts, simultaneously. In this paper, we propose a novel TKG embedding model which introduces the block term tensor decomposition and utilizes the core tensor and factor matrices to capture information presented by facts under distinct time granularities. By focusing on moments included in the time period and treating the discrete fact as a special case of the continuous fact, the model manages the processing of different types of facts in a unified manner. Besides, we explicitly design the static properties of entities and relations as well as their interactions to conform to reality. Experiments on 3 real datasets of different types verify the effectiveness of our proposed method compared with most state-of-the-art methods.

Application of Tucker Decomposition in Temperature Distribution Reconstruction

Constrained by cost, measuring conditions and excessive calculation, it is difficult to reconstruct a 3D real-time temperature field. For the purpose of solving these problems, a three-dimensional temperature distribution reconstruction algorithm based on Tucker decomposition algorithm is proposed. The Tucker decomposition algorithm is used to reduce the dimension of the measured data, and the processed core tensor is used for the temperature field reconstruction of sparse data. Theoretical analysis and simulations show that the proposed method is feasible; the overall optimization is realized by selecting the appropriate core tensor dimensions; and the reconstruction error is less than 3%. Results indicate that the proposed method can yield a reliable reconstruction solution and can be applied to real-time applications.

Low-Rank Tucker-2 Model for Multi-Subject fMRI Data Decomposition With Spatial Sparsity Constraint.

Tucker decomposition can provide an intuitive summary to understand brain function by decomposing multi-subject fMRI data into a core tensor and multiple factor matrices, and was mostly used to extract functional connectivity patterns across time/subjects using orthogonality constraints. However, these algorithms are unsuitable for extracting common spatial and temporal patterns across subjects due to distinct characteristics such as high-level noise. Motivated by a successful application of Tucker decomposition to image denoising and the intrinsic sparsity of spatial activations in fMRI, we propose a low-rank Tucker-2 model with spatial sparsity constraint to analyze multi-subject fMRI data. More precisely, we propose to impose a sparsity constraint on spatial maps by using an ℓp norm (0 < p ≤ 1), in addition to adding low-rank constraints on factor matrices via the Frobenius norm. We solve the constrained Tucker-2 model using alternating direction method of multipliers, and propose to update both sparsity and low-rank constrained spatial maps using half quadratic splitting. Moreover, we extract new spatial and temporal features in addition to subject-specific intensities from the core tensor, and use these features to classify multiple subjects. The results from both simulated and experimental fMRI data verify the improvement of the proposed method, compared with four related algorithms including robust Kronecker component analysis, Tucker decomposition with orthogonality constraints, canonical polyadic decomposition, and block term decomposition in extracting common spatial and temporal components across subjects. The spatial and temporal features extracted from the core tensor show promise for characterizing subjects within the same group of patients or healthy controls as well.

Spectral partitioning of large and sparse 3‐tensors using low‐rank tensor approximation

AbstractThe problem of partitioning a large and sparse tensor is considered, where the tensor consists of a sequence of adjacency matrices. Theory is developed that is a generalization of spectral graph partitioning. A best rank‐ approximation is computed for , and the partitioning is computed from the orthogonal matrices and the core tensor of the approximation. It is shown that if the tensor has a certain reducibility structure, then the solution of the best approximation problem exhibits the reducibility structure of the tensor. Further, if the tensor is close to being reducible, then still the solution of the exhibits the structure of the tensor. Numerical examples with synthetic data corroborate the theoretical results. Experiments with tensors from applications show that the method can be used to extract relevant information from large, sparse, and noisy data.

TensorGSEA: Detecting Differential Pathways in Type 2 Diabetes via Tensor-Based Data Reconstruction.

Detecting significant signaling pathways in disease progression highlights the dysfunctions and pathogenic mechanisms of complex disease development. Since tensor decomposition has been proven effective for multi-dimensional data representation and reconstruction, differences between original and tensor-processed data are expected to extract crucial information and differential indication. This paper provides a tensor-based gene set enrichment analysis, called tensorGSEA, based on a data reconstruction method to identify relevant significant pathways during disease development. As a proof-of-concept study, we identify the differential pathways of diabetes in rats. Specifically, we first arrange gene expression profiles of each documented pathway as tensors with three dimensions: genes, samples, and periods. Then we compress tensors into core tensors with lower ranks. The pathways with lower reconstruction rates are obtained after reconstructing gene expression profiles in another state via these cores. Thus, differences underlying pathways are extracted by cross-state data reconstruction between controls and diseases. The experiments reveal several critical pathways with diabetes-specific functions which otherwise cannot be identified by alternative methods. Our proposed tensorGSEA is efficient in evaluating pathways by achieving their empirical statistical significance, respectively. The classification experiments demonstrate that the selected pathways can be implemented as biomarkers to identify the diabetic state. The code of tensorGSEA is available at https://github.com/zhxr37/tensorGSEA .

Ultra-short-term wind speed and wind power forecast via selective Hankelization and low-rank tensor learning-based predictor

Accurate wind power or wind speed point forecast (WPF/WSF) can provide useful information for decision-makers to achieve a better energy management. This paper proposes an efficient univariate forecasting framework with a selective Hankelization (SH) technique and a low-rank tensor learning-based predictor (LRP). SH is applied to transform the original time series (e.g., wind data×time point) into a high-order tensor structure (e.g., wind data×similarity×time point). In SH, Hankelization introduces the additional dimension; similarity search (SS) collects the training samples sharing most similar features to the test sample; similarity rearranger (SR) reorders the fibres/vectors in tensors. The above three steps construct the translation invariance features in tensors’ 2D slices. With the tensor structure, the forecasting task is transferred to a higher dimension, where the proposed LRP performs efficiently. It applies Tucker decomposition (TD) for low-rank approximation and extracts low-rank core tensors for regression, which reduces information redundancy and computational cost. Then the low-rank tensor learning network (LRN), which implements a long short-term memory network (LSTM) with attention as an encoder and a multilayer perceptron (MLP) as a decoder, is designed for tensors regression in LRP. Such encoder–decoder network is used as the tensor-to-tensor learning network and can fit the correlation between slices well. Finally, experiments are carried out using wind speed/power data obtained from two datasets. The results demonstrate that the proposed method, compared to the mainstream global forecasting methods, improves the NMAE, NRMSE, and MAPE criteria by 22%, 25%, and 19% for WPF and by 9%, 11%, and 8% for WSF. It also outperforms some state-of-the-art local forecasting methods in terms of accuracy, which improves the three criteria by 8%, 11%, and 7% for WPF and by 3%, 7%, and 3% for WSF. In this process, the mapping to high dimension, the use of SS, the strategy of multi-step sampling and the architecture of the LRN all play positive and effective roles in improving the accuracy.

Tensor decomposition based on the potential low-rank and p-shrinkage generalized threshold algorithm for analyzing cancer multiomics data.

Tensor Robust Principal Component Analysis (TRPCA) has achieved promising results in the analysis of genomics data. However, the TRPCA model under the existing tensor singular value decomposition ([Formula: see text]-SVD) framework insufficiently extracts the potential low-rank structure of the data, resulting in suboptimal restored components. Simultaneously, the tensor nuclear norm (TNN) defined based on [Formula: see text]-SVD uses the same standard to handle various singular values. TNN ignores the difference of singular values, leading to the failure of the main information that needs to be well preserved. To preserve the heterogeneous structure in the low-rank information, we propose a novel TNN and extend it to the TRPCA model. Potential low-rank space may contain important information. We learn the low-rank structural information from the core tensor. The singular value space contains the association information between genes and cancers. The [Formula: see text]-shrinkage generalized threshold function is utilized to preserve the low-rank properties of larger singular values. The optimization problem is solved by the alternating direction method of the multiplier (ADMM) algorithm. Clustering and feature selection experiments are performed on the TCGA data set. The experimental results show that the proposed model is more promising than other state-of-the-art tensor decomposition methods.

Three-dimensional wind velocity reconstruction based on tensor decomposition and CFD data with experimental verification

Wind energy, which has many advantages, is in a stage of rapid development. However, because wind is sporadic and random, it is difficult to ensure a stable and efficient power supply, which poses risks to the security and stability of the power system. Therefore, research on short-term wind prediction is of great importance. Previous forecasting methods based on vectors or matrices have only been applied to wind velocity distributions in two-dimensional planes. If applied to multiple planes in three-dimensional (3D) space, these methods may not accurately reflect wind velocity distributions. To address this, we propose a novel method of wind forecasting: a tensor-based method that combines Tucker decomposition and computational fluid dynamics (CFD) to rapidly reconstruct 3D wind velocity distributions. A fourth-order wind velocity tensor database under three terrains is established by CFD simulation, then dimensionality reduction and feature extraction are carried out on the database by Tucker decomposition. The coefficient tensors obtained by decomposition are used to rapidly reconstruct 3D wind velocity distributions. Wind fields are successfully reconstructed with good accuracy for direction angles ranging from 0° to 180° and inlet speeds ranging from 0 to 33 m/s. The influences of core tensor dimension, the number and distribution of sensors, and noise on reconstruction error are discussed in the error analysis. Ultimately, the proposed method is verified by anemometer values from a wind tunnel experiment. The minimum relative reconstruction error is 1.79%. The experimental results show that the proposed method can accurately reconstruct 3D wind velocity distributions in wind fields and is an innovative method of short-term wind forecasting.

Low Rank Graph Regularization Embedding for 2D+3D Facial Expression Recognition

In this paper, a novel low rank graph regularization embedding for 3D facial expression recognition (LRGREFER) approach is proposed, in which the core tensor is utilized to characterize the low-rank attribute among the samples combined with the factor matrices with the graph regularization embedding. At first, a model based on a 4D tensor is constructed from the facial expression data. By Tucker decomposing the constructed 4D tensor, a resulting core tensor and factor matrices in different tensor modes are utilized to characterize the low-rankness among samples. Because of the loss of information during modelling the 4D tensor, the missing data from partly observed facial expression data are recovered by embedding the tensor completion. Finally, the proposed model is handled and solved by adopting the alternating direction method of multipliers (ADMM). Meanwhile, the classification prediction of facial expressions are implemented by Multi-class-SVM. Numerical experiments are conducted on BU-3DFE database. The experiment results have been verified that our proposed approach is more competitive.

Updating the Multilinear UTV Decomposition

Low multilinear rank approximation (LMLRA) is a basic tool for compressing a tensor into a more compact form, while preserving most of its information. An LMLRA of an <inline-formula><tex-math notation="LaTeX">$N$</tex-math></inline-formula>th order tensor consists of <inline-formula><tex-math notation="LaTeX">$N$</tex-math></inline-formula> factor matrices which are combined using a core tensor of smaller dimensions than the original tensor. When the tensor is non-static, for example when over time new tensor slices are being added along a certain mode, one can expect that the LMLRA of the tensor changes, but that an approximation of the new tensor can be derived from the previous one. By updating the factor matrices and the core tensor in an efficient way when a new tensor slice is added, this can indeed be achieved. In this paper, a new method is proposed to track the factor matrices in both the evolving and non-evolving modes, leading to efficient and accurate updates of the LMLRA. We track a truncated multilinear UTV decomposition (TMLUTVD), which is a tensor decomposition that is multilinear rank-revealing, yet less expensive to compute than the popular multilinear singular value decomposition. We derive accuracy bounds for the tracked TMLUTVD and give strategies to speed up its computation. Experiments show that the TMLUTVD tracking method works well in practice, even if the mode-<inline-formula><tex-math notation="LaTeX">$n$</tex-math></inline-formula> principal subspaces change drastically between updates. As an illustration of the possibilities of the technique, we extend the use of LMLRA in a compression step, prior to the computation of a canonical polyadic decomposition (CPD), to the setting of updating and tracking.

Rank determination in tensor factor model

Factor model is an appealing and effective analytic tool for high-dimensional time series, with a wide range of applications in economics, finance and statistics. This paper develops two criteria for the determination of the number of factors for tensor factor models where the signal part of an observed tensor time series assumes a Tucker decomposition with the core tensor as the factor tensor. The task is to determine the dimensions of the core tensor. One of the proposed criteria is similar to information based criteria of model selection, and the other is an extension of the approaches based on the ratios of consecutive eigenvalues often used in factor analysis for panel time series. Theoretically results, including sufficient conditions and convergence rates, are established. The results include the vector factor models as special cases, with an additional convergence rates. Simulation studies provide promising finite sample performance for the two criteria.

Low Rank Tucker Decomposition for 2D+3D Facial Expression Recognition

In this paper, a novel low rank Tucker decomposition for 3D facial expression recognition (LRTDFER) approach is proposed, in which the core tensor is utilized to characterize the low-rank attribute among the samples combined with the factor matrices with the graph regularization embedding. At first, a model based on a 4D tensor is constructed from the facial expression data. By Tucker decomposing the constructed 4D tensor, a resulting core tensor and factor matrices in different tensor modes are utilized to characterize the low-rankness among samples. Because of the loss of information during modelling the 4D tensor, the missing data from partly observed facial expression data are recovered by embedding the tensor completion. Finally, the proposed model is handled and solved by adopting the alternating direction method of multipliers (ADMM). Meanwhile, the classification prediction of facial expressions are implemented by Multi-class-SVM. Numerical experiments are conducted on BU-3DFE database. The experiment results have been verified that our proposed approach is more competitive.

SubTTD: DOA Estimation via Sub-Nyquist Tensor Train Decomposition

Conventional tensor direction-of-arrival (DOA) estimation methods for sparse arrays apply canonical polyadic decomposition (CPD) to the high-order coarray covariance tensor for retrieving angle information. However, due to the low convergence rate of CPD-based algorithms for high-order tensors, these methods suffer from a high computation cost. To address this issue, a sub-Nyquist tensor train decomposition (SubTTD)-based DOA estimation method is proposed for a three-dimensional (3-D) sparse array, where an augmented virtual array is derived from the sub-Nyquist tensor statistics. To reduce computational complexity of processing the 6-D coarray covariance tensor, the proposed SubTTD model efficiently decomposes it into a train of head matrix, 3-D core tensors, and tail matrix. Based on that, a core tensor decomposition and a change-of-basis transformation for the head matrix are designed to retrieve canonical polyadic factors of the coarray covariance tensor for DOA estimation. The computational efficiency of the proposed method is theoretically analyzed, and its effectiveness is verified via simulations.

Joint Factors and Rank Estimation for the Canonical Polyadic Decomposition Based on Convex Optimization

Estimating the minimal number of rank-1 tensors in the Canonical Polyadic Decomposition (CPD) known as the canonical rank is a challenging area of research. To address this problem, we propose a method based on convex optimization to jointly estimate the CP factors and the canonical rank called FARAC for Joint FActors and RAnk canonical estimation for the PolyAdiC Decomposition. We formulate the FARAC method as a convex optimization problem in which a sparse promoting constraint is added to the superdiagonal of the core tensor of the CPD while the frobenius norm of the offdiagonal terms is constrained to be bounded. We propose an alternated minimization strategy of the Lagrangien to solve the proposed optimization problem. Compared to the state-of-the-art method, FARAC shows very good performance in terms of accuracy of the rank estimation for large range of values of SNRs. In addition, FARAC can handle the case where the canonical rank exceeds one of the dimensions of the input tensor. Several numerical experiments are conducted in the presence of various levels of noise.

Automated Localization of Myocardial Infarction of Image-Based Multilead ECG Tensor With Tucker2 Decomposition

Myocardial infarction (MI) causes rapid and permanent damage to the heart muscle. Without timely diagnosis and treatment, it will deteriorate the myocardial structure and function. The precise localization of MI based on 12-lead electrocardiogram (ECG) signals still remains a great challenge.We thus present a novel algorithm for automatically localizing MI from multi-lead ECG signals. The image-based heartbeat tensorization establishes a third-order lead×time×amplitude tensor structure. This image tensor encompasses key information between leads and the correlation within the heartbeat, which are essential for the MI diagnosis. The Tucker2 decomposition based feature extractor automatically extracts the morphological core tensor of the image tensor. The morphological core tensor includes crucial information among three dimensions. Localization of MI is evaluated as a multi-class problem. We use the bagged decision tree for multi-class classification. The 12-lead ECG signals from the benchmark Physikalisch-Technische Bundesanstalt (PTB) database are employed to verify the applicability of the proposed algorithm. The PTB database includes normal ECG, 11 types of MI: anterior, anterior lateral, anterior septal, anterior septal lateral, inferior, inferior lateral, inferior posterior, inferior posterior lateral, lateral, posterior and posterior lateral. We demonstrated, with the morphological core features obtained from the image tensor, that 12 categories of ECG signals achieved an total accuracy of 99.67% and F1 score of 0.9997. The area under the receiver operating characteristic curves and precision-recall curves of each kind of ECG signal has been found to be more than 0.88. The proposed algorithm effectively realizes the classification of normal ECG and 11 categories of MI, and our approach of using 12-lead ECG signal herein holds great promise for helping the cardiologists localize MI.

High-Order Coupled Fully Connected Tensor Network Decomposition for Hyperspectral Image Super-Resolution

Hyperspectral image super-resolution addresses the problem of fusing a low-resolution hyperspectral image (LR-HSI) and a high-resolution multispectral image (HR-MSI) to produce a high-resolution hyperspectral image (HR-HSI). Tensor analysis has been proven to be an efficient method for hyperspectral image processing. However, the existing tensor-based methods of hyperspectral image super-resolution like the tensor train and tensor ring decomposition only establish an operation between adjacent two factors and are highly sensitive to the permutation of tensor modes, leading to an inadequate and inflexible representation. In this paper, we propose a novel method for hyperspectral image super-resolution by utilizing the specific properties of high-order tensors in fully-connected tensor network decomposition. The proposed method first tensorizes the target HR-HSI into a high-order tensor that has multiscale spatial structures. Then, a coupled fully-connected tensor network decomposition model is proposed to fuse the corresponding high-order tensors of LR-HSI and HR-MSI. Moreover, a weighted-graph regularization is imposed on the spectral core tensors to preserve spectral information. In the proposed model, the superiorities of the fully-connected tensor network decomposition lie in the outstanding capability for characterizing adequately the intrinsic correlations between any two modes of tensors and the essential invariance for transposition. Experimental results on three data sets show the effectiveness of the proposed approach as compared to other hyperspectral image super-resolution methods.

Polarization Image Demosaicking via Nonlocal Sparse Tensor Factorization

Division-of-focal-plane (DoFP) polarimeter provides a way for snapshot acquisition, making it available to simultaneously record polarization measurements at different orientations. This polarization imaging system has gained more attention in the last few years and is promising to be used in the fields of computer vision and remote sensing. However, this system suffers from the degradation of spatial resolution. To reconstruct polarization information at full resolution, polarization image demosaicking is indispensable. To address polarization image demosaicking issue while preserving the essential structure of polarization data, a sparse tensor factorization-based model is proposed. For a target cube, its similar cubes are first grouped together as a tensor. Then, its compact dictionary and sparse core tensor are learned by factorizing the tensor using sparse coding. Moreover, the correlation among different polarization orientations and the nonlocal self-similarity are adopted to boost the performance. Experimental results on synthetic and real-world data demonstrate that our proposed model outperforms several state-of-the-art methods in terms of both quantitative measurements and visual quality.

Nonintrusive Model Order Reduction for Cross-Diffusion Systems

In this paper, we investigate tensor based nonintrusive reduced-order models (ROMs) for parametric cross-diffusion equations. The full-order model (FOM) consists of ordinary differential equations (ODEs) in matrix or tensor form resulting from finite-difference discretization of the differential operators by taking the advantage of Kronecker structure. The matrix/tensor differential equations are integrated in time with the implicit-explicit (IMEX) Euler method. The reduced bases, relying on a finite sample set of parameter values, are constructed in form of a two-level approach by applying higher-order singular value decomposition (HOSVD) to the space-time snapshots in tensor form, which leads to a large amount of computational and memory savings. The nonintrusive reduced approximations for an arbitrary parameter value are obtained through tensor product of the reduced basis by the parameter dependent core tensor that contains the reduced coefficients. The reduced coefficients for new parameter values are computed using radial basis function (RBF) interpolation. The efficiency of the proposed method is illustrated through numerical experiments for two-dimensional Schnakenberg and three-dimensional Brusselator cross-diffusion equations. The spatiotemporal patterns are accurately predicted by the reduced-order models with speed-up factors of orders two and three over the full-order models.