Tensor-based Method 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.

Tensor-Based Data-Driven Identification of Partial Differential Equations

Abstract We present a tensor-based method for model selection which identifies the unknown partial differential equation that governs a dynamical system using only spatiotemporal measurements. The method circumvents a disadvantage of standard matrix-based methods which typically have large storage consumption. Using a recently developed multidimensional approximation of nonlinear dynamical systems (MANDy), we collect the nonlinear and partial derivative terms of the measured data and construct a low-rank dictionary tensor in the tensor-train (TT) format. A tensor-based linear regression problem is then built, which balances the learning accuracy, model complexity, and computational efficiency. An algebraic expression of the unknown equations can be extracted. Numerical results are demonstrated on datasets generated by the wave equation, the Burgers' equation, and a few parametric partial differential equations (PDEs).

A tensor decomposition scheme for EEG-based diagnosis of mild cognitive impairment

Mild Cognitive Impairment (MCI) is the primary stage of acute Alzheimer's disease, and early detection is crucial for the person and those around him. It is difficult to recognize since this mild stage does not have clear clinical signs, and its symptoms are between normal aging and severe dementia. Here, we propose a tensor decomposition-based scheme for automatically diagnosing MCI using Electroencephalogram (EEG) signals. A new projection is proposed, which preserves the spatial information of the electrodes to construct a data tensor. Then, using parallel factor analysis (PARAFAC) tensor decomposition, the features are extracted, and a support vector machine (SVM) is used to discriminate MCI from normal subjects. The proposed scheme was tested on two different datasets. The results showed that the tensor-based method outperformed conventional methods in diagnosing MCI with an average classification accuracy of 93.96% and 78.65% for the first and second datasets, respectively. Therefore, it seems that maintaining the spatial topology of the signals plays a vital role in the processing of EEG signals.

An Approach for Using a Tensor-Based Method for Mobility-User Pattern Determining

Modern mobile networks exhibit a complex heterogeneous structure. To enhance the Quality of Service (QoS) in these networks, intelligent control mechanisms should be implemented. These functions are based on the processing of large amounts of data and feature extraction. One such feature is information about user mobility. However, directly determining user mobility remains challenging. To address this issue, this study proposes an approach based on multi-linear data processing. The user mobility is proposed to determine, using the multi-linear data, about the changing of the Signal-to-Interference-plus-Noise-Ratio (SINR). SINR varies individually for each user over time, relative to the network’s base stations. It is natural to represent these data as a tensor. A tensor-based preprocessing step employing Canonical Polyadic Decomposition (CPD) is proposed to extract user mobility information and reduce the data volume. In the next step, using the DBSCAN algorithm, users are clustered according to their mobility patterns. Subsequently, users are clustered based on their mobility patterns using the DBSCAN algorithm. The proposed approach is evaluated utilizing data from Network Simulator 3 (NS-3), which simulates a portion of the mobile network. The results of processing these data using the proposed method demonstrate superior performance in determining user mobility.

Synchrotron-based X-ray phase contrast imaging of transmural cardiac tissue in patients treated for advanced heart failure

Abstract Background Cardiac imaging is essential for identifying structural changes in advanced heart failure (HF) and understanding underlying pathophysiology. Synchrotron radiation-based X-ray phase contrast imaging (X-PCI) is a non-destructive imaging modality that can provide high resolution three-dimensional (3D) visualisation of cardiac tissue on the microstructural level. Thus far, such analyses including the quantification of myocyte aggregates ranging from the epicardium to the endocardium, have been confined to animal models and human fetal tissue. We aimed to explore the feasibility of using X-PCI to quantify and compare structural organisation and orientation of myocyte aggregates in the myocardium across different advanced HF aetiologies. Methods and design Four adult patients were included - two receiving a left ventricular assist device (LVAD), and two undergoing heart transplantation (HTx). Aetiology of advanced HF in both the LVAD and HTx group was ischaemic heart disease and dilated cardiomyopathy (DCM), respectively. Transmural tissue samples were obtained by left ventricular apical coring (LVAD) and from the explanted hearts (HTx). The tissue specimens were imaged by X-PCI at the TOMCAT beamline using an effective pixel size of 5.8 µm. Orientation of aggregates of cardiomyocytes was assessed using a structure tensor-based method and morphological parameters were derived - the helical angle (HA), or the angle between the circumferential left ventricular plane and myocyte aggregates, and fractional anisotropy (FA), the degree of anisotropy or disorganisation of the local myocardium. Results The general patient characteristics are shown in Table 1. X-PCI enabled 3D visualisation of transmural myocardial tissue samples showing differences in organisation of myocardial structure in different HF aetiologies (Figure 1). The epicardial adipose layer was thicker in DCM, paired with a thinner epicardial myocyte layer (yellow arrows). Furthermore, the epi-to-endocardial HA transition was clearly visualized in ICM (double arrows), whereas there was clear disruption of the HA gradient in DCM samples – a finding further quantified via FA (green arrows). Conclusion X-PCI allows non-destructive advanced imaging of ex-vivo tissue. By harnessing the power of digital 3D imaging on a cellular level, it provides a novel insight to differential information on tissue organisation in different aetiologies of advanced heart failure.Table 1Figure 1

TenGAN: adversarially generating multiplex tensor graphs

In this work, we explore multiplex graph (networks with different types of edges) generation with deep generative models. We discuss some of the challenges associated with multiplex graph generation that make it a more difficult problem than traditional graph generation. We propose TenGAN, the first neural network for multiplex graph generation, which greatly reduces the number of parameters required for multiplex graph generation. We also propose 3 different criteria for evaluating the quality of generated graphs: a graph-attribute-based, a classifier-based, and a tensor-based method. We evaluate its performance on 4 datasets and show that it generally performs better than other existing statistical multiplex graph generative models. We also adapt HGEN, an existing deep generative model for heterogeneous information networks, to work for multiplex graphs and show that our method generally performs better.

Higher Order Time-Frequency Domain Tensor-Based Method for the Detection of Atrial Abnormalities Using 12-Lead ECG Signals

Higher Order Time-Frequency Domain Tensor-Based Method for the Detection of Atrial Abnormalities Using 12-Lead ECG Signals

Tensor-based flow reconstruction from optimally located sensor measurements

Reconstructing high-resolution flow fields from sparse measurements is a major challenge in fluid dynamics. Existing methods often vectorize the flow by stacking different spatial directions on top of each other, hence confounding the information encoded in different dimensions. Here, we introduce a tensor-based sensor placement and flow reconstruction method which retains and exploits the inherent multidimensionality of the flow. We derive estimates for the flow reconstruction error, storage requirements and computational cost of our method. We show, with examples, that our tensor-based method is significantly more accurate than similar vectorized methods. Furthermore, the variance of the error is smaller when using our tensor-based method. While the computational cost of our method is comparable to similar vectorized methods, it reduces the storage cost by several orders of magnitude. The reduced storage cost becomes even more pronounced as the dimension of the flow increases. We demonstrate the efficacy of our method on three examples: a chaotic Kolmogorov flow, in situ and satellite measurements of the global sea surface temperature and three-dimensional unsteady simulated flow around a marine research vessel.

Target Parameter Estimation Algorithm Based on Real-Valued HOSVD for Bistatic FDA-MIMO Radar

Since there is a frequency offset between each adjacent antenna of FDA radar, there exists angle-range two-dimensional dependence in the transmitter. For bistatic FDA-multiple input multiple output (MIMO) radar, range-direction of departure (DOD)-direction of arrival (DOA) information is coupled in transmitting the steering vector. How to decouple the three information has become the focus of research. Aiming at the issue of target parameter estimation of bistatic FDA-MIMO radar, a real-valued parameter estimation algorithm based on high-order-singular value decomposition (HOSVD) is developed. Firstly, for decoupling DOD and range in transmitter, it is necessary to divide the transmitter into subarrays. Then, the forward–backward averaging and unitary transformation techniques are utilized to convert complex-valued data into real-valued data. The signal subspace is obtained by HOSVD, and the two-dimensional spatial spectral function is constructed. Secondly, the dimension of spatial spectrum is reduced by the Lagrange algorithm, so that it is only related to DOA, and the DOA estimation is obtained. Then the frequency increment between subarrays is used to decouple the DOD and range information, and eliminate the phase ambiguity at the same time. Finally, the DOD and range estimation automatically matched with DOA estimation are obtained. The proposed algorithm uses the multidimensional structure of high-dimensional data to promote performance. Meanwhile, the proposed real-valued tensor-based method can effectively cut down the computing time. Simulation results verify the high efficiency of the developed method.

An in silico osmotic pressure approach allows characterization of pressure-area isotherms of lipid monolayers at low molecular areas.

Surface pressure-area isotherms of lipid monolayers at the air-water interface provide essential information about the structure and mechanical behaviour of lipid membranes. These curves can be readily obtained through Langmuir trough measurements and, as such, have been collected for decades in the field of membrane biochemistry. However, it is still challenging to directly observe and understand nanoscopic features of monolayers through such experiments, and molecular dynamics (MD) simulations are generally used to provide a molecular view of such interfaces. In MD simulations, the surface pressure-area (Π-A) isotherms are generally computed using the Kirkwood-Irving formula, that relies on the evaluation of the pressure tensor. This approach, however, has intrinsic limitations when the molecular area in the monolayer is low (typically < 60 Å2 per lipid). Recently, an alternative method to compute Π-A isotherms of surfactants, based on the calculation of the three-dimensional osmotic pressure via the implementation of semipermeable barriers was proposed. In this work, we investigate the feasibility of this approach for long-chain surfactants such as phospholipids. We identify some discrepancies between the computed values and experimental results, and we propose a semi-empirical correction based on the molecular structure of the surfactants at the monolayer interface. To validate the potential of this new approach, we simulate several phosphatidylcholine and phosphatidylethanolamine lipids at various temperatures using all-atom and coarse-grained force fields, and we compute the corresponding Π-A isotherms. Our results show that the Π-A isotherms obtained using the new method are in very good agreement with experiments and far superior to the canonical pressure tensor-based method at low molecular areas. This corrected osmotic pressure method allows for accurate characterization of the molecular packing in monolayers in various physical phases.

Riemannian conjugate gradient descent method for fixed multi rank third-order tensor completion

The goal of tensor completion is to fill in missing entries of a partially known tensor under a low-rank constraint. In this paper, we study low rank third-order tensor completion problems by using Riemannian optimization methods on the smooth manifold. Here the tensor rank is defined to be a multi-rank under the Discrete Cosine Transform-related transform tensor-tensor product. With suitable incoherence conditions on the underlying low multi-rank rank tensor, we show that the proposed Riemannian optimization method is guaranteed to converge to the underlying low multi-rank tensor with a high probability. Number of sampling entries required for convergence are also derived. Numerical examples of synthetic data and real data sets are reported to demonstrate that the performance of the proposed method is better than that of tensor-based method using the Tucker-rank model in terms of computational time, and that of the matrix-based completion method in terms of number of sampling entries.

Segmentation of intrinsically very low contrast magnetic resonance brain images using tensor-based DTI registration

BackgroundIn patients with hypomyelinating leukodystrophies, contrast of T1-weighted brain MRI is very low due to the lack of myelin, preventing a reliable segmentation. In diffusion tensor images the contrast is higher, thanks to anisotropy and orientation of white matter (WM) tracts. We aimed to develop and assess a tensor-guided atlas-based segmentation method suitable for segmentation of very low contrast images. Methods17 control subjects (mean age 8.0 yrs (SD 8.0)) and 27 subjects with hypomyelinating leukodystrophies (mean age 10.7 yrs (SD 10.2)) were included. DTI and 3D T1 images were segmented using a DTI-TK tensor-guided IIT-atlas-based segmentation method. For the control subjects, these segmentations were compared with a conventional segmentation of their 3D T1-weighted images. A qualitative visual assessment and a quantitative assessment using DTI metrics was performed to assess the patient segmentations. ResultsIn control subjects, the tensor-based method performed as can be expected for atlas-based segmentation methods, with Dice coefficients of 0.65, 0.72, 0.81 and 0.86 for cortical grey matter (GM), WM, deep grey matter (DGM), and thalamus, respectively. In patients with hypomyelination the visual assessment showed anatomically adequate segmentations. All tissue-specific DTI metrics differed between patients and controls. Patients with hypomyelination had reduced FA and increased mean, axial and radial diffusivities, not only in total WM, but also in the corticospinal tracts, optic radiations and thalamus. ConclusionEven in the absence of normal myelin, the presence and direction of axons allowed tensor-based registration and thereby atlas-based segmentation. We showed the applicability of the segmentation method in the context of quantitative MRI, allowing for whole-brain or regional tissue-specific and tract-specific analyses of very low contrast images.

Developing Tensor-Based Common and Special Feature Analysis for Comprehensive Monitoring of Complex Batch Processes

To overcome the limitation of unfolding-based methods and handle the multiple data set and limited data problems in the complex processes, such as multigrade batch processes, a novel tensor-based common and special feature extraction method and a comprehensive monitoring framework are proposed. In the proposed method, the uneven-length three-dimensional data are directly analyzed by the comprehensive tensor-based method without unfolding. To handle the multiple data set modeling problem, the tensor-based common feature extraction methods are first proposed to obtain the common features shared among different grades. The special features are sequentially determined by conducting tensor principal component analysis (PCA) on the residuals of each grade. The data are thus divided into common, special, and residual subspaces. Three monitoring statistics are established respectively in each subspace for online fault detection. The merits and effectiveness of the proposed method are demonstrated by an injection molding process with both even-length and uneven-length data in comparison with traditional methods.

Spatially Smoothed Tensor-Based Method for Bistatic Co-Prime MIMO Radar With Hole-Free Sum-Difference Co-Array

In this paper, we investigate the problem of joint direction of departure (DoD) and direction of arrival (DoA) estimation with bistatic MIMO radar. Specifically, a weaved co-prime array (WCA) bistatic MIMO (WCA-bMIMO) radar is proposed to produce a hole-free sum-difference co-array, which enables to offer significantly increased degrees of freedom (DoFs) and enlarged array aperture. Moreover, the two-dimensional spatial smoothing method is introduced to construct a smoothed received signal, which helps to tackle the rank deduction problem rooted in the equivalent one-snapshot received signal from the sum-difference co-array. Subsequently, the joint DoD and DoA estimates can be obtained by extracting the three-way tensor model. Furthermore, co-array Cramer-Rao Bound (CRB) for the bistatic co-prime MIMO radar is provided. Finally, the superiorities of the proposed bistatic co-prime MIMO radar in accuracy and resolution are proved by numerical simulation results.

Sparse Nonnegative Tensor Factorization and Completion With Noisy Observations

In this paper, we study the sparse nonnegative tensor factorization and completion problem from partial and noisy observations for third-order tensors. Because of sparsity and nonnegativity, the underlying tensor is decomposed into the tensor-tensor product of one sparse nonnegative tensor and one nonnegative tensor. We propose to minimize the sum of the maximum likelihood estimation for the observations with nonnegativity constraints and the tensor <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\ell _{0}$ </tex-math></inline-formula> norm for the sparse factor. We show that the error bounds of the estimator of the proposed model can be established under general noise observations. The detailed error bounds under specific noise distributions including additive Gaussian noise, additive Laplace noise, and Poisson observations can be derived. Moreover, the minimax lower bounds are shown to be matched with the established upper bounds up to a logarithmic factor of the sizes of the underlying tensor. These theoretical results for tensors are better than those obtained for matrices, and this illustrates the advantage of the use of nonnegative sparse tensor models for completion and denoising. Numerical experiments are provided to validate the superiority of the proposed tensor-based method compared with the matrix-based approach.

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.

Total variation regularized nonlocal low-rank tensor train for spectral compressive imaging

In this paper, we consider the reconstruction problem of hyperspectral images (HSIs) based on a new type of imaging system termed as coded aperture snapshot spectral imaging (CASSI). CASSI compresses multiple spectral bands of HSI into a single snapshot measurement which enjoys low cost, low bandwidth, and high-speed sensing rate. Despite the promising applications of CASSI systems, the bottleneck lies in the reconstruction algorithms. The existing CASSI reconstruction methods could not fully exploit the underlying structures of HSIs, and thereby demonstrate unsatisfactory reconstruction quality. To remedy this issue, we propose in this paper a novel tensor-based method by modeling two intrinsic HSI priors, namely nonlocal low-rank tensor train (NLTT) and 3-D weighted total variation (3DWTV). Taking into consideration that the snapshot measurement and targeted HSI should share similar spatial structures, we use the snapshot measurement to help group nonlocal spatially similar cubes within the unknown HSI to form 4-D tensors. And the NLTT prior employed on these 4-D tensors can effectively learn the correlations among the spatial, spectral and nonlocal modes thanks to the well-balanced matricization scheme of TT rank. Meanwhile, we employ the 3DWTV prior to characterize the local smoothness of the spatial and spectral modes. The resulting model can be solved efficiently by using the alternative direction method of multipliers (ADMM). Extensive experiments on both simulated and real CASSI datasets validate the improved reconstruction performance of our method over several other competing ones.

Spatiotemporal Road Traffic Anomaly Detection: A Tensor-Based Approach

The increased development of urban areas results in a larger number of vehicles on the road network, leading to traffic congestion, which often leads to potentially dangerous situations that can be described as anomalies. The tensor-based methods emerged only recently in applications related to traffic anomaly detection. They outperform other models regarding simultaneously capturing spatial and temporal components, which are of immense importance in traffic dataset analysis. This paper presents a tensor-based method for extracting the spatiotemporal road traffic patterns represented with the speed transition matrices, with the goal of anomaly detection. A novel anomaly detection approach is presented, which relies on computing the center of mass of the observed traffic patterns. The method was evaluated on a large road traffic dataset and was able to detect the most anomalous parts of the urban road network. By analyzing spatial and temporal components of the most anomalous traffic patterns, sources of anomalies can be identified. Results were validated using the extracted domain knowledge from the Highway Capacity Manual. The anomaly detection model achieved a precision score of 92.88%. Therefore, this method finds its usages for safety experts in detecting potentially dangerous road segments, urban traffic planners, and routing applications.

A Robust Tensor-Based Submodule Clustering for Imaging Data Using l12 Regularization and Simultaneous Noise Recovery via Sparse and Low Rank Decomposition Approach.

The massive generation of data, which includes images and videos, has made data management, analysis, information extraction difficult in recent years. To gather relevant information, this large amount of data needs to be grouped. Real-life data may be noise corrupted during data collection or transmission, and the majority of them are unlabeled, allowing for the use of robust unsupervised clustering techniques. Traditional clustering techniques, which vectorize the images, are unable to keep the geometrical structure of the images. Hence, a robust tensor-based submodule clustering method based on regularization with improved clustering capability is formulated. The induced tensor nuclear norm (TNN), integrated into the proposed method, offers better low rankness while retaining the self-expressiveness property of submodules. Unlike existing methods, the proposed method employs a simultaneous noise removal technique by twisting the lateral image slices of the input data tensor into frontal slices and eliminates the noise content in each image, using the principles of the sparse and low rank decomposition technique. Experiments are carried out over three datasets with varying amounts of sparse, Gaussian and salt and pepper noise. The experimental results demonstrate the superior performance of the proposed method over the existing state-of-the-art methods.

Bearing early fault detection and degradation tracking based on support tensor data description with feature tensor

In view of easily interfered properties of bearing incipient fault signal, it is beneficial to mine more useful information from vibration signal for bearing state monitoring. For exploring local information and utilizing correlation between the characteristic of different frequency bands, feature tensor based on Wavelet Packet (WP) decomposition is introduced to characterize bearing fault state as reflected by vibration signal. Then the bearing state monitoring can be transformed as a one-class classification problem in tensor space, in which the abnormal samples represent the fault states and the normal samples mean the health states of the bearings, respectively. Combining with Support Tensor Data Description (STDD), an early fault detection method is proposed by employing feature tensor correlating with bearing fault directly. To further quantify the fault degree of bearing, a tensor-based anomaly index (AIt) is presented by taking advantage of the distance between feature tensor sample to spherical surface generated by STDD in tensor space. Experimental results of bearing verify that the proposed tensor-based fault detection method is effective for bearing early fault detection, and AIt is sensitive to bearing operation state and therefore suitable for fault degradation tracking. Further experiments show that the tensor-based method is robust to noise and small sample size problem.