Four-dimensional Tensor Research Articles

Spatial transition tensor of single cells

Spatial transcriptomics and messenger RNA splicing encode extensive spatiotemporal information for cell states and transitions. The current lineage-inference methods either lack spatial dynamics for state transition or cannot capture different dynamics associated with multiple cell states and transition paths. Here we present spatial transition tensor (STT), a method that uses messenger RNA splicing and spatial transcriptomes through a multiscale dynamical model to characterize multistability in space. By learning a four-dimensional transition tensor and spatial-constrained random walk, STT reconstructs cell-state-specific dynamics and spatial state transitions via both short-time local tensor streamlines between cells and long-time transition paths among attractors. Benchmarking and applications of STT on several transcriptome datasets via multiple technologies on epithelial–mesenchymal transitions, blood development, spatially resolved mouse brain and chicken heart development, indicate STT’s capability in recovering cell-state-specific dynamics and their associated genes not seen using existing methods. Overall, STT provides a consistent multiscale description of single-cell transcriptome data across multiple spatiotemporal scales.

Robust Tensor Hypercontraction of the Particle-Particle Ladder Term in Equation-of-Motion Coupled Cluster Theory.

One method of representing a high-rank tensor as a (hyper-)product of lower-rank tensors is the tensor hypercontraction (THC) method of Hohenstein et al. This strategy has been found to be useful for reducing the polynomial scaling of coupled-cluster methods by representation of a four-dimensional tensor of electron-repulsion integrals in terms of five two-dimensional matrices. Pierce et al. have already shown that the application of a robust form of THC to the particle-particle ladder (PPL) term reduces the cost of this term in couple-cluster singles and doubles (CCSD) from to with negligible errors in energy with respect to the density-fitted variant. In this work, we have implemented the least-squares variant of THC (LS-THC) which does not require a nonlinear tensor factorization, including the robust form (R-LS-THC), for the calculation of the excitation and electron attachment energies using equation-of-motion coupled cluster methods EOMEE-CCSD and EOMEA-CCSD, respectively. We have benchmarked the effect of the R-LS-THC-PPL approximation on excitation energies using the comprehensive QUEST database and the accuracy of electron attachment energies using the NAB22 database. We find that errors on the order of 1 meV are achievable with a reduction in total calculation time of approximately 5 ×.

Relativistic constitutive modeling of inelastic deformation of continua moving in space-time

Since Einstein introduced the theory of relativity, several scientific observations have proven that it thoroughly approximates the motion of materials in space-time. Thus, efforts have been made to expand classical continuum mechanics to relativistic continuum models to consider relativistic effects. However, most models consider only the elastic range without accounting for inelastic deformation. A relativistic inelasticity model should provide objective measures of the inelastic deformation for different observers with nonzero relative velocities over a wide speed range, even when the moving speed is close to the speed of light. This study presents a mathematical modeling structure of the relativistic constitutive equations of the inelastic deformation of a material moving over a wide speed range to provide objective measures for observers. In this model, the deformation tensors of classical mechanics are expanded to four-dimensional tensors (referred to relativistic Cauchy–Green deformation tensors) in space-time, based on which constitutive equations of inelastic deformation are introduced. The four-dimensional tensors are objective about the homogeneous Lorentz transformation in the Minkowski space-time, and the material dissipation, determined employing the proposed modeling of inelastic deformation, satisfies the second law of thermodynamics. In addition, an example illustrates the application of this theory by explaining the physical meaning of modeling. Finally, it is also demonstrated that the proposed relativistic inelasticity model collapses into a classical inelasticity model when the speed of motion is much slower than the speed of light.

A dynamic modeling method using channel-selection convolutional neural network: A case study of NOx emission

A novel channel-selection convolutional neural network (CS-CNN) is proposed to predict NOx emission from coal-fired boilers under steady-state and transient load conditions. First, a new channel-selection convolutional layer (CS-CL) is presented to replace regular convolutional layer (RCL). The CS-CL evaluates the channel importance of the input variables, selects the Top-C important channels and releases the hyperparameters of the remaining low-importance channels, thus contributing to maximize the utilization of the parameter resources of the model. The advantages of using CS-CLs are the preservation of the great majority of manipulated variables involved in combustion control among the input variables and the prevention of the model overfitting problem due to the redundancy of input variables. Second, a sliding window-based preprocessing method is applied to the historical data of the boiler which is divided into four-dimensional (4D) tensors. Then, comparative tests are performed on long short-term memory (LSTM) model, baseline CNN and CS-CNN using the historical data of a 670 MW boiler. The results of tests showed that CS-CNN has higher prediction performance. Finally, in order to increase the interpretability of the deep learning black box model, this study analyzes the working mechanism of the CS-CNN through ablation analysis and visualization of model parameters.

Traffic Target Location Estimation Based on Tensor Decomposition in Intelligent Transportation System

As the safety problems and economic losses caused by traffic accidents are becoming more and more serious, intelligent transportation system (ITS) came into being. After the outbreak of COVID-19, how to achieve effective traffic scheduling and macro command under less contact has attracted more attention. Therefore, the location estimation of traffic objectives is a key issue. In the developed framework, for the target parameter estimation in traffic, frequency diversity array multiple-input multiple-output (FDA-MIMO) radar is introduced into ITS, and tensor decomposition is used to process transportation big data (TBD) to improve the real-time performance of target location estimation. Unfortunately, spatial colored noise and array gain-phase error will affect the performance of FDA-MIMO radar in ITS. An algorithm that can solve the angle-range estimation problem of FDA-MIMO radar in the co-existence of array gain-phase error and spatial colored noise is proposed. Firstly, the four-dimensional tensor is constructed by using the temporal un-correlation of colored noise. Therefore, the influence of colored noise in ITS is removed. Secondly, the direction matrix containing target information is obtained by parallel factor (PARAFAC) decomposition. For the array gain-phase error, the optimization problem is constructed, and the Lagrange multiplier is employed to calculate the optimal solution. The effect of gain-phase error is eliminated by utilizing the optimal solution and the direction matrices. Finally, the location information of motor vehicle is achieved by calculating the solution of least square (LS) fitting. The developed scheme can achieve the location information of motor vehicles in the co-existence of array gain-phase error and spatial colored noise. Comprehensive numerical experiments illustrate that the developed scheme in ITS can efficiently obtain the location information of motor vehicles.

Infrared Dim and Small Target Detection Based on Superpixel Segmentation and Spatiotemporal Cluster 4D Fully-Connected Tensor Network Decomposition

The detection of infrared dim and small targets in complex backgrounds is very challenging because of the low signal-to-noise ratio of targets and the drastic change in background. Low-rank sparse decomposition based on the structural characteristics of infrared images has attracted the attention of many scholars because of its good interpretability. In order to improve the sensitivity of sliding window size, insufficient utilization of time series information, and inaccurate tensor rank in existing methods, a four-dimensional tensor model based on superpixel segmentation and statistical clustering is proposed for infrared dim and small target detection (ISTD). First, the idea of superpixel segmentation is proposed to eliminate the dependence of the algorithm on selection of the sliding window size. Second, based on the improved structure tensor theory, the image pixels are statistically clustered into three types: corner region, flat region, and edge region, and are assigned different weights to reduce the influence of background edges. Next, in order to better use spatiotemporal correlation, a Four-Dimensional Fully-Connected Tensor Network (4D-FCTN) model is proposed in which 3D patches with the same feature types are rearranged into the same group to form a four-dimensional tensor. Finally, the FCTN decomposition method is used to decompose the clustered tensor into low-dimensional tensors, with the alternating direction multiplier method (ADMM) used to decompose the low-rank background part and sparse target part. We validated our model across various datasets, employing cutting-edge methodologies to assess its effectiveness in terms of detection precision and reduction of background interference. A comparative analysis corroborated the superiority of our proposed approach over prevailing techniques.

A note on the Kaluza–Klein theory

We show that the Kaluza–Klein (KK) theory contains a fundamental problem: The four-dimensional metric tensor and the electromagnetic potential vector assumed in the KK theory belong to four-dimensional vector spaces that are not integrable in general, resulting that the four-dimensional physical variables and the corresponding field equations derived from the five-dimensional Einstein field equation (i.e. the four-dimensional Einstein field equation and the Maxwell equations) are not defined on a four-dimensional submanifold. That is, the four-dimensional spacetime assumed in the KK theory does not exist. No satisfactory solutions are found within the KK formalism. Perhaps the best approach to fix the problem is giving up the KK theory and looking for a new unified scheme for gravitational and electromagnetic interactions in the framework of a spacetime with extra dimensions, as having already been explored in some literature.

Attention-deep reinforcement learning jointly beamforming based on tensor decomposition for RIS-assisted V2X mmWave massive MIMO system

To achieve a green and sustainable wireless communication network, the reconfigurable intelligent surface (RIS) technology has emerged as an emerging technology. The RIS-assisted mmWave massive MIMO vehicle-to-everything (V2X) communications can support enhanced V2X applications for connected and automated vehicles. The design of RIS-assisted mmWave V2X communications is though not exempt of challenges as a result of fast time-varying propagation and highly dynamic vehicular networks and topologies. To address some of these challenges, we propose an attention-deep reinforcement learning jointly beamforming based on tensor decomposition for RIS-assisted mmWave massive MIMO system to improving the safety and traffic efficiency of cooperative automated driving. The received signal of the vehicle is decomposed into four-dimensional tensors and study the channel, frequency and time attention (CFTA) algorithm using the low order tensor characteristics of mmWave channels. This algorithm can extract the frequency and time characteristics of the channel, and obtain the millimeter wave characteristic channel between the corresponding base station (BS) and the vehicles. The problem of jointly optimization of beamforming matrix at the BS and the phase shift matrix on the RIS is constructed and solved using deep reinforcement learning (DRL). Unlike traditional alternating optimization methods, the integration of DRL techniques into the optimal design of the system enables the observation of immediate rewards, learning from the environment and improving it to obtain optimal solutions to high-dimensional jointly beamforming optimization problems ultimately. The simulation results show that the proposed algorithm has meaningful performance compared with other methods.

Research on multi-context aware recommendation methods based on tensor factorization

Compared to the traditional recommender systems, context-aware recommender systems are more in line with actual application contexts. However, the existing researches are mostly focused on single context-aware recommendation, such as time-aware recommendation or location-aware recommendation, and lack of in-depth research on multi-context-aware recommendation. Therefore, we proposed a recommendation method of high-order tensor factorization based on multi-context-aware. First, on the basis of analyzing the influence of context on users’ interest preferences, the sensitivity of users to multiple contexts was detected using statistical methods. For context-sensitive users, four-dimensional tensors and feature matrices used to solve data sparsity were constructed based on rating matrix and situational information. And then the stochastic gradient descent algorithm was used for iterative calculation to fill in missing data values and carry out parameter optimization. For context-insensitive users, we used matrix factorization to predict users’ interest preferences. Finally, we tested and validated our method on a multi-context-aware movie dataset, and the experimental results show that the proposed method could effectively reduce the prediction error and improve the recommendation quality.

Missing data recovery of wind speed in wind farms: A spatial-temporal tensor decomposition approach

Missing data recovery plays a critical role in improving the data quality of wind speed in wind farms, and numerous methods have been proposed to address this issue. However, most of them suffer from the inability to fully use the information of known data, and thus, poor performance of recovery is usually achieved. In this paper, we propose a missing data recovery method based on spatial-temporal tensor decomposition. The proposed method rearranges the whole data based on discrete wavelet transform to construct a four-dimensional tensor of “site × week × scale × hour” for representing the spatial and temporal correlation of wind speed. A completeness tensor is estimated to impute missing data based on Tucker decomposition and the nonlinear conjugate gradient algorithm. The proposed method not only inherits the advantages of imputation methods based on the matrix pattern but also well mines the spatial and temporal inherent correlation of wind speed. Wind speed data of a wind farm are used to verify the effectiveness of the proposed method. The results show that the proposed method recovers missing data with much smaller mean absolute error and root mean square error and requires less effort for recovering missing data of fragmented or continuously, compared with the traditional methods.

Exact analytical vacuum solutions of [formula omitted]-gravity model depending on two variables

In this paper we consider the metric power-law f(R)∼Rn-gravity model for the four-dimensional metric tensor depending on two coordinates. We obtain exact analytical vacuum solutions for different values of n. These solutions contain both non-stationary configurations of the travelling wave type and stationary ones, in particular, depending on one radial variable.

High Dimensional Convolutional Neural Network for EEG Connectivity-Based Diagnosis of ADHD.

Attention-deficit/hyperactivity disorder (ADHD) is a common neurodevelopmental disorder in children and adults and its early detection is effective in the successful treatment of children. Electroencephalography (EEG) has been widely used for classifying ADHD and normal children. In recent years, deep learning leads to more accurate classification. This study aims to adapt convolutional neural networks (CNNs) for classifying ADHD and normal children based on the connectivity measure of their EEG signals. In this experimental study, the dataset consisted of 61 ADHD and 60 normal children from which 13021 epochs were extracted as input for model training and evaluation. Synchronization likelihood (SL) and wavelet coherence (WC) were considered connectivity measures. The neighborhood between EEG channels was arranged in a two-dimensional matrix for better representation. Four-dimensional (4D) and six-dimensional (6D) connectivity tensors were composed as model inputs. Two architectures were developed, one 4D and 6D CNN for SL and WC-based diagnosis of ADHD, respectively. A 5-fold cross-validation was utilized to assess developed models. The average accuracy of 98.56% for 4D CNN and 98.85% for 6D CNN in epoch-based classification were obtained. In the case of subject-based classification, the accuracy was 99.17% for both models. Based on the evaluation metrics of the proposed models, ADHD children can be diagnosed and ADHD and normal children can be successfully distinguished.

On four-way CP model estimation efficiency

The latent structure of four-dimensional tensors can be investigated by means of the four-way CANDECOMP/PARAFAC model. This technique is seldom used because its estimating design is challenging from an algorithmic and interpretational standpoint. Parameter estimation with a least-squares approach can be computationally costly, especially under difficult conditions such as factor collinearity and model over-specification. In this work, we implement a 4th-order extension of the efficient trilinear procedure INT-2 to tackle estimating setbacks and test it in a simulation study.

Hyperspectral and multispectral data fusion via nonlocal low-rank learning

Low-spatial-resolution hyperspectral images have rich spectral information and poor spatial resolution, whereas high-spatial-resolution multispectral images have high spatial resolution and limited spectral information. Current methods cannot simultaneously consider the structure of spatial and spectral domains of an hyperspectral image cube. To solve this problem, we propose a fusion model based on spatial nonlocal similarity and low-rank prior learning. The proposed method first extracts a series of fixed-size, full-band tensor blocks, select their corresponding reference blocks to search similar full-band blocks in its search window, list similar blocks to form tensor groups, and then use a four-dimensional tensor structure to extract local full-band blocks. Next, a low-rank regularization term is introduced to the fusion model. Finally, the fusion problem is transformed to a low-rank regularization optimization problem, which is solved by the alternating direction multiplier method. Comparison with state-of-the-art methods demonstrates the method’s effectiveness.

Skeleton-based Action Recognition with Two-Branch Graph Convolutional Networks

Graph convolution is a popular technique for action recognition based on human skeleton data. Due to the fact that human skeleton data can be treated as a graph in three dimensions, Graph Convolutional Networks (GCN) represent the input data as a graph structure to perform the recognition task, and thus numerous approaches based on GCN to recognize actions have achieved great results. In spite of the input data being structured as a four-dimensional tensor in GCN, it still can not fully exploit the contained rich action-related information. Therefore, we proposed a new model that the three-dimensional skeleton data is put into both the Convolutional Neural Network (CNN) and the Graph Convolutional Neural Network branches to perform feature extraction in the spatiotemporal dimension separately, then the output information is fused for prediction. Given the richness of time-domain information characteristics, feature extraction is enhanced by increasing the model’s depth. After the pooling layer and the fully connected layer, we concatenate the outputs at the ends of the graph data stream and the convolution data stream to obtain the network’s final output. Finally, the prediction results can be obtained via the SoftMax layer. On the Kinetic 400 dataset, our suggested model outperforms Benchmark STGCN in terms of accuracy. The experiment results indicate that the proposed novel model successfully increases the generalization ability and classification performance for action recognition.

A Tensor Train Compression Scheme for Remote Volume-surface Integral Equation Interactions

We propose a memory compression scheme for the coupling matrices appearing in volume-surface integral equation formulations. When there is some distance between the surface and the volumetric scatterers, the low-rank properties of the coupling matrix, allow us to reshape it into a set of four-dimensional tensors, which can be heavily compressed with the tensor train decomposition. The associated matrix-vector products can be rapidly performed with the aid of a graphical processing unit. We achieved a compression of more than 8 thousand times with a relative error around 1e – 5, for the calculation of the electromagnetic field generated by a radiofrequency coil inside an object at 1 mm voxel isotropic resolution. In this case, the matrix-vector product could be executed in less than a second.

Induced equation of state for the universe epochs constrained by the hubble parameter

We present a five dimensional cosmological metric that reveals a four-dimensional energy-momentum tensor. We analyse three cases for the resulting field equations: null, positive and negative cosmological constant Λ. For the case with null cosmological constant, we obtain a solution able to describe the radiation-dominated era of the universe. The positive five-dimensional cosmological constant case yields a bounce cosmological model. In the negative Λ case, the scale factor for the line element is obtained as a(t)=c5sinh(23|Λ|t), where c5 is a constant. This solution can remarkably describe not only the late-time cosmic acceleration but also the non-accelerated stages of the cosmic expansion, namely the radiation and matter dominated epochs in a continuous form. We obtain an analytical equation of state capable of describing the different epochs of the universe and it reads as p=ω(z)ρ with ω(z)=13[1−41+c54(1+z)4], with p being the pressure of the universe, ρ its density and z is the redshift. In our model the constant c5 is related to the Hubble constant as H0=|Λ|6coth[arcsinh(1c5)2] and this satisfactorily fits the observational data for the low redshift sample of the experimental measurements of the Hubble parameter, which results in H0=72.2−5.5+5.3 km s−1 Mpc−1 and c5=0.600−0.058+0.061.

Transfer Urban Human Mobility via POI Embedding over Multiple Cities

Rapidly developing location acquisition technologies provide a powerful tool for understanding and predicting human mobility in cities, which is very significant for urban planning, traffic regulation, and emergency management. However, with the existing methodologies, it is still difficult to accurately predict millions of peoples’ mobility in a large urban area such as Tokyo, Shanghai, and Hong Kong, especially when collected data used for model training are often limited to a small portion of the total population. Obviously, human activities in city are closely linked with point-of-interest (POI) information, which can reflect the semantic meaning of human mobility. This motivates us to fuse human mobility data and city POI data to improve the prediction performance with limited training data, but current fusion technologies can hardly handle these two heterogeneous data. Therefore, we propose a unique POI-embedding mechanism, that aggregates the regional POIs by categories to generate an artificial POI-image for each urban grid and enriches each trajectory snippet to a four-dimensional tensor in an analogous manner to a short video. Then, we design a deep learning architecture combining CNN with LSTM to simultaneously capture both the spatiotemporal and geographical information from the enriched trajectories. Furthermore, transfer learning is employed to transfer mobility knowledge from one city to another, so that we can fully utilize other cities’ data to train a stronger model for the target city with only limited data available. Finally, we achieve satisfactory performance of human mobility prediction at the citywide level using a limited amount of trajectories as training data, which has been validated over five urban areas of different types and scales.

A Transfer and Deep Learning-Based Method for Online Frequency Stability Assessment and Control

Fast and accurate prediction and control of power system dynamic frequency after disturbance is essential to enhance power system stability. Machine learning methods have great potential in harnessing data for online application with accurate predictions. This paper proposes a two-stage novel transfer and deep learning-based method to predict power system dynamic frequency after disturbance and provide optimal event-based load shedding strategy to maintain system frequency. The proposed deep learning model combines convolutional neural network (CNN) and long short-term memory (LSTM) network to harness both spatial and temporal measurements in the input data, through a four-dimensional (4-D) tensor input construction process including, 1) capture system network topology information and critical measurements from different time intervals; 2) compute a multi-dimensional electric distance matrix and reduce to a 2-D plane which can describe the system nodal distribution; 3) construct 3-D tensors based on state variables at different sample times; and 4) integrate into 4-D tensor inputs. Moreover, a transfer learning process is employed to overcome the challenge of insufficient data and operating condition changes in real power systems for new prediction tasks. Simulation results in IEEE 118-bus system verify that the CNN-LSTM method not only greatly improves the timeliness of online frequency control, but also presents good accuracy and effectiveness. Test cases in the New England 39-bus system and the South Carolina 500-bus system validate that the transfer learning process can provide accurate results even with insufficient training data.

Integrated fusion framework based on semicoupled sparse tensor factorization for spatio-temporal–spectral fusion of remote sensing images

Remote sensing image fusion is considered a cost effective method for handling the tradeoff between the spatial, temporal and spectral resolutions of current satellite systems. However, most current fusion methods concentrate on fusing images in two domains among the spatial, temporal and spectral domains, and a few efforts have been made to comprehensively explore the relationships of spatio-temporal–spectral features. In this study, we propose a novel integrated spatio-temporal–spectral fusion framework based on semicoupled sparse tensor factorization to generate synthesized frequent high-spectral and high-spatial resolution images by blending multisource observations. Specifically, the proposed method regards the desired high spatio-temporal–spectral resolution images as a four-dimensional tensor and formulates the integrated fusion problem as the estimation of the core tensor and the dictionary along each mode. The high-spectral correlation across the spectral domain and the high self-similarity (redundancy) features in the spatial and temporal domains are jointly exploited using the low dimensional and sparse core tensors. In addition, assuming that the sparse coefficients in the core tensors across the observed and desired image spaces are not strictly the same, we formulate the estimation of the core tensor and the dictionaries as a semicoupled sparse tensor factorization of available heterogeneous spatial, spectral and temporal remote sensing observations. Finally, the proposed method can exploit the multicomplementary spatial, temporal and spectral information of any combination of remote sensing data based on this single unified model. Experiments on multiple data types, including spatio-spectral, spatio-temporal, and spatio-temporal–spectral data fusion, demonstrate the effectiveness and efficiency of the proposed method.