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Tensor-ring (TR) decomposition was recently studied and applied for low-rank tensor completion due to its powerful representation ability of high-order tensors. However, most of the existing TR-based methods tend to suffer from deterioration when the selected rank is larger than the true one. To address this issue, this article proposes a new low-rank sparse TR completion method by imposing the Frobenius norm regularization on its latent space. Specifically, we theoretically establish that the proposed method is capable of exploiting the low rankness and Kronecker-basis-representation (KBR)-based sparsity of the target tensor using the Frobenius norm of latent TR-cores. We optimize the proposed TR completion by block coordinate descent (BCD) algorithm and design a modified TR decomposition for the initialization of this algorithm. Extensive experimental results on synthetic data and visual data have demonstrated that the proposed method is able to achieve better results compared to the conventional TR-based completion methods and other state-of-the-art methods and, meanwhile, is quite robust even if the selected TR-rank increases.

Following traumatic brain injury (TBI), cerebral metabolic dysfunction, characterized by an elevated cerebral microdialysis (CMD) lactate/pyruvate (LP) ratio, is associated with poor outcome. However, the exact pathophysiological mechanisms underlying this association are not entirely established. In this pre-planned analysis of the BIOmarkers of AXonal injury after Traumatic Brain Injury (BIO-AX-TBI) prospective study, we investigated any associations of LP ratio with brain structure volume change rates at 1 year. Fourteen subjects underwent acute-phase (0-96 h post-TBI) CMD monitoring and had longitudinal magnetic resonance imaging (MRI) quantification of brain volume loss between the subacute phase (14 days to 6 weeks) and 1 year after TBI, recalculated as an annual rate. On average, CMD showed an elevated (>25) LP ratio (31 [interquartile range (IQR) 24-34]), indicating acute cerebral metabolic dysfunction. Annualized whole brain and total gray matter (GM) volume change rates were abnormally reduced (-3.2% [-9.3 to -2.2] and -1.9% [-4.4 to 1.7], respectively). Reduced annualized total GM volume correlated significantly with elevated CMD LP ratio (Spearman's rho = -0.68, p-value = 0.01) and low CMD glucose (rho = 0.66, p-value = 0.01). After adjusting for age, admission Glasgow Coma Scale (GCS) score and CT Marshall score, CMD LP ratio remained strongly associated with 1-year total GM volume change rate (p < 0.001; multi-variable analysis). No relationship was found between WM volume changes and CMD metabolites. We demonstrate a strong association between acute post-traumatic cerebral metabolic dysfunction and 1-year gray matter atrophy, reinforcing the role of CMD LP ratio as an early biomarker of poor long-term recovery after TBI.

Background The imaging of brain tumours has significantly improved with the use of advanced magnetic resonance (MR) techniques like diffusion tensor imaging (DTI). This study was conducted to analyse the utility of DTI-derived tensor metrics in the evaluation of intracranial gliomas with histopathological correlation and further adoption of these image-data analyses in clinical setting. Methods A total of 50 patients with suspected diagnosis of intracranial gliomas underwent DTI along with conventional MR examination. The study correlated various DTI parameters in the enhancing part of the tumour and the peritumoral region with the histopathological grades of the intracranial gliomas. Results The study revealed higher values of Cl (linear anisotropy), Cp (planar anisotropy), AD (axial diffusivity), FA (fractional anisotropy) and RA (relative anisotropy) and lower values of Cs (spherical anisotropy), MD (mean diffusivity) and RD (radial diffusivity) in the enhancing part of the tumour in case of high-grade gliomas. However, in the peritumoral region, the values of Cl, Cp, AD, FA and RA were less whereas values of Cs, MD and RD were more in high-grade gliomas than in the low-grade gliomas. The various cutoff values of these DTI-derived tensor metrics were found to be statistically significant. Conclusion DTI-derived tensor metrics can be a valuable tool in differentiation between high-grade and low-grade gliomas which might be accepted in clinical practice in near future.

Spectrum map is an essential tool for a range of emerging applications of 5G and 6G networks. Despite the great efforts that have been put on the construction of spectrum maps, access to accurate and valid spectrum data in dynamically changing environments emphasizes the need for more advanced solutions. To this end, the idea of spectrum map prediction is introduced. In this paper, we address the problem of spectrum map prediction from incomplete and corrupted historical spectrum observations. We propose three techniques to solve the problem. First, we combine the spectrum map with prediction functionalities so as to offer a huge potential in dynamically changing environments. Second, by fully exploiting the inherent spatial-temporal-spectral structures of the spectrum data and the sparsity of anomalies and missing data, we model the spectrum map as a 3rd-order spectrum tensor and formulate the spectrum map prediction problem as a low-rank tensor completion problem. Third, we design a robust online spectrum map prediction (ROSMP) algorithm based on the alternating direction minimization method. Finally, extensive numerical evaluations using a real spectrum measurement dataset confirm the efficacy and efficiency of ROSMP and show the superiority of ROSMP over the baselines.

Notice that it is difficult and expensive to implement global network measurements to obtain network distance, a feasible idea is to predict unknown distances by introducing network coordinates with limited network measurements. The existing solutions always represent the unknown network distances in a rather unique number. However, research and applications indicate that the real network distances are hard to be accurately figured out and changes subtly in an interval over time with the dynamic network environments. Accordingly, this paper proposes a tensor-based network distance prediction (TNDP) approach to represent network distance with confidence intervals, by exploiting the random distance tensor and distributed matrix factorization. With a small set of network measurements among the nodes selected randomly, a distance matrix tensor has been established and factorized into the product of two location matrixes with the adaptive SGD-based learning solution. By introducing the important training determinants, including weight matrix, regularization coefficient, and minibatch gradient descent with the exponential decay rates, the unknown distances among nodes can be accurately inferred in the forms of confidence intervals, with quick convergence and less overfitting. Extensive experimental simulations demonstrate that TNDP is superior to other approaches in terms of accuracy for network distance predictio

With the bloom of intelligent devices over the world, a large scale of user's trajectory data are collected. How to mine valuable rules from these data and provide services for the industrial community has become an urgent problem. In this paper, we propose a multi-modal prediction system to infer users' mobility pattern embedded in heterogeneous data from Cyber-Physical-Social space. The highlight is the retrospective multivariate multi-step transition tensor chain model, which decomposes a large scale of multivariate multi-step transition tensor into a series of small scale transition tensors (sub-transition tensors) with the tensor maximum likelihood estimation (TMLE) method. Then that one can solve the stationary probability distribution with the small scale sub-transition tensors so as to highly reduce the computation and storage cost. At the same time, the tensor maximum likelihood estimation method avoids the over-fitting of multivariate multi-step transition tensor, the proposed model improves the performance of prediction systems.

Tensor singular value decomposition (t-SVD) has recently become increasingly popular for tensor recovery under partial and/or corrupted observations. However, the existing t -SVD-based methods neither make use of a rank prior nor provide an accurate rank estimation (RE), which would limit their recovery performance. From the practical perspective, the tensor RE problem is nontrivial and difficult to solve. In this article, we, therefore, aim to determine the correct rank of an intrinsic low-rank tensor from corrupted observations based on t-SVD and further improve recovery results with the estimated rank. Specifically, we first induce the equivalence of the tensor nuclear norm (TNN) of a tensor and its f -diagonal tensor. We then simultaneously minimize the reconstruction error and TNN of the f -diagonal tensor, leading to RE. Subsequently, we relax our model by removing the TNN regularizer to improve the recovery performance. Furthermore, we consider more general cases in the presence of missing data and/or gross corruptions by proposing robust tensor principal component analysis and robust tensor completion with RE. The robust methods can achieve successful recovery by refining the models with correct estimated ranks. Experimental results show that the proposed methods outperform the state-of-the-art methods with significant improvements.

Tensor decompositions, such as CANDECOMP/PARAFAC (CP), are widely used in a variety of applications, such as chemometrics, signal processing, and machine learning. A broadly used method for computing such decompositions relies on the Alternating Least Squares (ALS) algorithm. When the number of components is small, regardless of its implementation, ALS exhibits low arithmetic intensity, which severely hinders its performance and makes GPU offloading ineffective. We observe that, in practice, experts often have to compute multiple decompositions of the same tensor, each with a small number of components (typically fewer than 20), to ultimately find the best ones to use for the application at hand. In this article, we illustrate how multiple decompositions of the same tensor can be fused together at the algorithmic level to increase the arithmetic intensity. Therefore, it becomes possible to make efficient use of GPUs for further speedups; at the same time, the technique is compatible with many enhancements typically used in ALS, such as line search, extrapolation, and non-negativity constraints. We introduce the Concurrent ALS algorithm and library, which offers an interface to MATLAB, and a mechanism to effectively deal with the issue that decompositions complete at different times. Experimental results on artificial and real datasets demonstrate a shorter time to completion due to increased arithmetic intensity.

Non-negative tensor factorization has been shown a practical solution to automatically discover phenotypes from the electronic health records (EHR) with minimal human supervision. Such methods generally require an input tensor describing the inter-modal interactions to be pre-established; however, the correspondence between different modalities (e.g., correspondence between medications and diagnoses) can often be missing in practice. Although heuristic methods can be applied to estimate them, they inevitably introduce errors, and leads to sub-optimal phenotype quality. This is particularly important for patients with complex health conditions (e.g., in critical care) as multiple diagnoses and medications are simultaneously present in the records. To alleviate this problem and discover phenotypes from EHR with unobserved inter-modal correspondence, we propose the collective hidden interaction tensor factorization (cHITF) to infer the correspondence between multiple modalities jointly with the phenotype discovery. We assume that the observed matrix for each modality is marginalization of the unobserved inter-modal correspondence, which are reconstructed by maximizing the likelihood of the observed matrices. Extensive experiments conducted on the real-world MIMIC-III dataset demonstrate that cHITF effectively infers clinically meaningful inter-modal correspondence, discovers phenotypes that are more clinically relevant and diverse, and achieves better predictive performance compared with a number of state-of-the-art computational phenotyping models., Comment: Accepted by IEEE Transactions on Knowledge and Data Engineering (TKDE)

In learning tasks with high order correlations, the low-rank approximation of the regression coefficient tensor has become increasingly important. Tensor ring can capture more correlation information among tensor networks. However, its optimal rank is generally unknown and needs to be tuned from multiple combinations. To address the issue, we propose a novel tensor regression framework with a group sparsity constraint on latent factors for tensor ring rank estimation. Specifically, the proposed group sparsity term constrained matrix factorization problem is first proved to be equivalent to a better approximation of matrix rank, namely Schatten- ${1/2}$ quasi-norm. Extending it into tensor, the tensor ring rank can be inferred during the learning process to balance the prediction error and the model complexity. Besides, a total variation term is introduced to enhance the local consistency of the predicted response, which is useful for reducing the adverse effects of random noise. Experiments on the simulation dataset show that the proposed method can exactly obtain the tensor ring rank, and the effectiveness and robustness of the proposed algorithm is further verified on a real dataset for human motion capture tasks.

Heterogeneous Information Network(HIN) collecitve classification studies the problem of predicting labels for one type of nodes in a HIN which contains multiple types of nodes multiple types of links among them. Previous studies have revealed that exploiting relative importance of links is quite useful to improve node classification performance as connected nodes tend to have similar labels. Most existing approaches exploit the relative importance of links either by directly counting the number of connections among nodes or by learning the weight of each type of link from labeled data only. However, these approaches either neglect the importance of types of links to the class labels or may lead to overfitting problem. We propose a Tensor-based Markov chain (T-Mark) approach, which is able to automatically and simultaneously predict the labels for unlabeled nodes and give the relative importance of types of links that actually improve the classification accuracy. Specifically, we build two tensor equations by using the HIN and features of nodes from both labeled and unlabeled data. A Markov chain-based model is proposed and it is solved by an iterative process to obtain the stationary distributions. Theoretical analyses of the existence and uniqueness of such probability distributions are given. Extensive experimental results demonstrate that T-Mark is able to achieve superior performance in the comparison and obtain reasonable relative importance of links.

In this paper, starting from the algorithm, a performance- and energy-efficient 3D structure or shape of the Tensor Processing Engine (TPE) for CNN acceleration is systematically searched and evaluated. An optimal accelerator's shape maximizes the number of concurrent MAC operations per clock cycle while minimizes the number of redundant operations. The proposed 3D vector-parallel TPE architecture with an optimal shape can be very efficiently used for considerable CNN acceleration. Due to inter-block image data independency, it is possible to use multiple of such TPEs for the additional CNN acceleration. Moreover, it was shown that proposed TPE can also be uniformly used for acceleration of the different CNN models such as VGG, ResNet, YOLO and SSD.

In this paper, we focused on the study of anisotropic strange stars with anisotropic matter distribution in the framework of f ( T , T ) = α T ( r ) 2 + β T ( r ) model, where α , β are arbitrary constants, T and T represent the torsion scalar and trace of energy–momentum tensor, respectively. This model has interesting cosmological properties, in addition to providing an effective dark energy sector explanation (Harko et al., 2014). In order to achieve our goal, we considered the Krori-Barua space–time and utilize the off-diagonal tetrad components for deriving the corresponding set of field equations. We calculate the relation for the evaluation of effective density, effective radial and tangential pressure using the EoS of strange star i.e., MIT bag model p r ( r ) = 1 3 [ ρ ( r ) − 4 B g ] . The unknown constants appearing in the solution have been evaluated by using the conventional matching of interior and exterior space time. To check the feasibility of our proposed model we discuss the anisotropy, energy condition, TOV forces, abreu condition, EoS parameters, adiabatic index, mass function, compactness and redshift for different compact stars models namely, P S R J 1416 − 2230 , 4 U 1608 − 52 , C e n X − 3 , E X O 1785 − 248 and S M C X − 1 .

BACKGROUND: Shortening of tensor fasciae latae is one factor that causes a functional leg length discrepancy. A shoe lift has been used to correct the compensatory posture resulting from the discrepancy. Despite the potential therapeutic benefit of a shoe lift, the mechanism by which it exerts its effect is unclear. OBJECTIVE: To investigate the effect of a shoe lift on tensor fasciae latae length during standing with an artificial functional leg length discrepancy using ultrasonic shear wave elastography. METHODS: Twenty-two healthy individuals performed static standing under three conditions: drop of the pelvis and flexion of the leg resulting from fixing in the hip abduction position using a hip orthosis (functional leg length discrepancy condition); drop of the pelvis by the orthosis, but no flexion of the leg due to a shoe lift (shoe lift condition); and normal bilateral standing condition. The shear elastic modulus of tensor fasciae latae was calculated using ultrasonic shear wave elastography. RESULTS: The shear elastic modulus was significantly lower in the functional leg length discrepancy condition than in the shoe lift and normal conditions (p= 0.038). CONCLUSIONS: Using a shoe lift for the functional leg length discrepancy can result in a functional hip position that elongates tensor fasciae latae.

Dynamic relationships are frequently encountered in big data and services computing-related applications, like dynamic data of user-side QoS in Web services. They are modeled into a high-dimensional and sparse (HiDS) tensor, which contain rich knowledge regarding temporal patterns. A non-negative latent factorization of tensors (NLFT) model is very effective in extracting such patterns from an HiDS tensor. However, it commonly suffers from overfitting with improper regularization schemes. To address this issue, this study investigates NLFT models with diversified regularization schemes. Six regularized NLFT models, i.e., L2, L1, elastic net, log, dropout, and swish-regularized ones, are proposed and carefully investigated. Moreover, owing to their diversified regularization designs, they possess strong model diversity to achieve an effective ensemble. Empirical studies on HiDS QoS tensors from real applications demonstrate that compared with state-of-the-art models, the proposed ones better describe the temporal patterns hidden in an HiDS tensor, thereby achieving significantly higher prediction accuracy for missing data. Moreover, their ensemble further outperforms each of them in terms of prediction accuracy for missing QoS data.

Human-oriented and machine-generated data in cyber-physical-social systems are often complicated graph-structured. Graph-powered learning methods are conducive to discovering valuable knowledge from large scale graph data and improving decision making processes. However, due to the neglect of diverse relations among things, most existing knowledge reasoning studies are inherently flawed and inefficient in processing the heterogeneous graphs with high-order connectivity. Tensor, as a powerful and effective tool to model high-level semantic interactions between various things, can provide high-order IoT graph with new perspectives and possibilities. Therefore, this paper innovatively proposes a collaborative Artificial Intelligence of Things data analysis and application framework based on Boolean tensors, which supports the expression and fusion of heterogeneous graph and ultimately promotes the AI processing. In this context, we focus on developing an incremental Boolean tensor factorization (IBTF) approach for efficient knowledge reasoning to meet the requirements of real-time and high level quality demands for intelligent services. To the best of our knowledge, we are the first to do this work. More concretely, we present factors update and binary features merge algorithms for the integrated graph tensors to avoid numerous repeated calculations of historical data. Experimental results on general synthetic data sets demonstrate that the IBTF approach proposed in this paper guarantees nearly equal approximate accuracy while reducing execution time by dozens and even more of times. Furthermore, experimental evaluations and interpretability analysis on real-world datasets verify the practicality of the proposed framework and approach.

The key to hyperspectral anomaly detection is to effectively distinguish anomalies from the background, especially in the case that background is complex and anomalies are weak. Hyperspectral imagery (HSI) as an image-spectrum merging cube data can be intrinsically represented as a third-order tensor that integrates spectral information and spatial information. In this article, a prior-based tensor approximation (PTA) is proposed for hyperspectral anomaly detection, in which HSI is decomposed into a background tensor and an anomaly tensor. In the background tensor, a low-rank prior is incorporated into spectral dimension by truncated nuclear norm regularization, and a piecewise-smooth prior on spatial dimension can be embedded by a linear total variation-norm regularization. For anomaly tensor, it is unfolded along spectral dimension coupled with spatial group sparse prior that can be represented by the l2,1-norm regularization. In the designed method, all the priors are integrated into a unified convex framework, and the anomalies can be finally determined by the anomaly tensor. Experimental results validated on several real hyperspectral data sets demonstrate that the proposed algorithm outperforms some state-of-the-art anomaly detection methods.

The first-order resonant Raman spectra of monolayer MoS2 are calculated under the circularly polarized photoexcitation. The anomalously nonzero Raman intensity of the in-plane E mode under the Z ¯ ( σ + σ + ) Z or Z ¯ ( σ − σ − ) Z geometry, which goes against the conventional selection rule, appears under some circumstances when optical absorption occurs at some special reciprocal points between the zone-center Γ and the zone-edge-center M points. At that moment, the valley selectivity to the circular polarization is lifted. The analysis shows that the anomalous Raman intensity of the E mode for the same circularly polarized incident and scattered light is consistent with the pseudo-angular-momentum conservation law. The calculated E Raman tensor of monolayer MoS2 is found to vary with laser energy. The two diagonal terms of the Raman tensor change their signs from mutually opposite to the same when the relative intensity of the in-plane E mode to the out-of-plane A 1 ′ mode increases, indicating the increasingly important role played by the Frolich-type electron-phonon interaction over the deformation potential. Our study may shed new light on the understanding of the novel electron-photon process and assist in the design of new type of optoelectronic devices.

Deep image prior (DIP), which uses a deep convolutional network (ConvNet) structure as an image prior, has attracted wide attention in computer vision and machine learning. DIP empirically shows the effectiveness of the ConvNet structures for various image restoration applications. However, why the DIP works so well is still unknown. In addition, the reason why the convolution operation is useful in image reconstruction, or image enhancement is not very clear. This study tackles this ambiguity of ConvNet/DIP by proposing an interpretable approach that divides the convolution into ``delay embedding'' and ``transformation'' (i.e., encoder-decoder). Our approach is a simple, but essential, image/tensor modeling method that is closely related to self-similarity. The proposed method is called manifold modeling in embedded space (MMES) since it is implemented using a denoising autoencoder in combination with a multiway delay-embedding transform. In spite of its simplicity, MMES can obtain quite similar results to DIP on image/tensor completion, super-resolution, deconvolution, and denoising. In addition, MMES is proven to be competitive with DIP, as shown in our experiments. These results can also facilitate interpretation/characterization of DIP from the perspective of a ``low-dimensional patch-manifold prior.''

In recent years, Krylov subspace methods based on the tensor format have demonstrated their superiority over classical ones for handling various kinds of tensor equations. In this paper, according to the efficient computational cost of the classical Simpler GMRES method (SGMRES), we adopt the tensor format of this method, which is called SGMRES−BTF, for solving the Sylvester tensor equations. This paper is divided into two parts describing the tensor format of two types of acceleration approaches to tackle some serious problems of the restarted methods GMRES−BTF and SGMRES−BTF, such as the “stalling” and “alternating phenomenon”. In the first part of this two-part work, the truncated Loose Simpler GMRES based on the tensor format (LSGMRES−BTF) is introduced, which is an acceleration approach that aims to construct an augmented tensor Krylov subspace by means of approximation error tensors. Moreover, a detailed study is carried out on the connection between the values of sequential and skip angles and the convergence behavior of both SGMRES−BTF and truncated LSGMRES−BTF. In the second part of this paper, an acceleration approach based on the idea of inner-outer iteration in the truncated version of the generalized conjugate residual with inner orthogonalization (GCRO) method is developed. In this method, SGMRES−BTF is applied in the inner iteration, and the generalized conjugate residual based on the tensor format (GCR−BTF) method is used in the outer iteration. Numerical experiments show that SGMRES−BTF achieves an appropriate performance compared with GMRES−BTF. In addition, the numerical results reveal the high potential of the presented accelerating strategies to deal with Sylvester tensor equations.

In the second part of this two-part work, another accelerator method based on the tensor format is established for solving the Sylvester tensor equations. This acceleration approach, which is called SGCRO−BTF, is based on the idea of inner-outer iteration used in the generalized conjugate residual with inner orthogonalization (GCRO) method. In SGCRO−BTF, the Simpler GMRES method based on the tensor format (SGMRES−BTF) is applied to the inner iteration, and the generalized conjugate residual based on the tensor format (GCR−BTF) method is used in the outer iteration. Furthermore, SGCRO−BTF seeks an approximate solution over a tensor subspace spanned by the approximation error tensors produced during the previous outer iterations of SGCRO−BTF and a tensor Krylov subspace constructed by the inner iteration. In order to reduce the computational storage in the outer iteration, the truncated version of SGCRO−BTF is presented, in which only some of the last approximation error tensors are kept and will be then added to the tensor Krylov subspace in order to obtain a new search subspace. Finally, the proposed methods are tested on a set of experiments and compared to some conventional and state-of-the-art Krylov subspace methods based on the tensor format. Experimental results indicate that the truncated version of SGCRO−BTF is particularly effective for solving the Sylvester tensor equations.

Low-rank modeling has achieved great success in visual data completion. However, the low-rank assumption of original visual data may be in approximate mode, which leads to suboptimality for the recovery of underlying details, especially when the missing rate is extremely high. In this paper, we go further by providing a detailed analysis about the rank distributions in Hankel structured and clustered cases, and figure out both non-local similarity and patch-based structuralization play a positive role. This motivates us to develop a new Hankel low-rank tensor recovery method that is competent to truthfully capture the underlying details with sacrifice of slightly more computational burden. First, benefiting from the correlation of different spectral bands and the smoothness of local spatial neighborhood, we divide the visual data into overlapping 3D patches and group the similar ones into individual clusters exploring the non-local similarity. Second, the 3D patches are individually mapped to the structured Hankel tensors for better revealing low-rank property of the image. Finally, we solve the tensor completion model via the well-known alternating direction method of multiplier (ADMM) optimization algorithm. Due to the fact that size expansion happens inevitably in Hankelization operation, we further propose a fast randomized skinny tensor singular value decomposition (rst-SVD) to accelerate the periteration running efficiency. Extensive experimental results on real world datasets verify the superiority of our method compared to the state-of-the-art visual inpainting approaches.

Diffusion kurtosis imaging (DKI) has been shown to be valuable in a wide range of neuroscientific and clinical applications. However, reliable estimation of DKI tensors is often compromised by noise, especially for the kurtosis tensor (KT). Here, we propose a joint denoising and estimating framework that integrates multiple sources of prior information, including nonlocal structural self-similarity (NSS), local spatial smoothness (LSS), physical relevance (PR) of the DKI model, and noise characteristics of magnitude diffusion MRI (dMRI) images for improved estimation of DKI tensors. The local and nonlocal spatial smoothing constraints are complementary to each other, making the proposed framework highly effective in reducing the noise fluctuations on DKI tensors, especially KT. As an additional refinement, we propose to impose a physically relevant constraint within our joint denoising and estimation framework. We further adopt the first-moment noise-corrected fitting model (M1NCM) to remove the noncentral χ-distribution noise bias. The effectiveness of integrating multiple sources of priors into the joint framework is verified by comparing the proposed M1NCM-NSS-LSS-PR method with various versions of M1NCM-based estimators and two state-of-the-art methods. Results show that the proposed method outperformed the compared methods in simulations and in-vivo dMRI datasets of both spatially stationary and nonstationary noise distributions. The in-vivo experiments also show that the proposed M1NCM-NSS-LSS-PR method was robust to the number of diffusion directions.

We propose a novel architecture to efficiently perform sparse tensor decomposition (SpTD). As the generalization of vectors and matrices, tensors are widely used to process high-dimensional data. SpTD is not only an emerging tensor analysis technique but also an effective tool to reduce the storage and computation costs of tensors. However, conventional general-purpose processors are inefficient to perform SpTD, mainly due to: i) variable sparsity degree and flexible buffer size requirement; ii) difficulties of fusing multiple execution kernels to pursue better performance. For domain-specific accelerator designers on the other hand, the diversity of decomposition algorithms is also an important problem that must be considered. We propose a unified abstraction for SpTD algorithms and design a specialized accelerator. First, we formulate two types of core kernels (SpLrMM and LrSampling) that serve as a standard form to fit a broad range of SpTD algorithms. Second, we design a sparse tensor engine (STE) to efficiently perform SpTD. STE uses a processing element (PE)-interactive architecture where PEs can be flexibly grouped together via Network-on-Chip to share the buffer capacity, bandwidth, and compute resources. We evaluate our accelerator with extensive experiments, and it achieves an average speedup of 45x over CPU and 29x over GPU.

In this paper we mainly investigate the Cauchy problem of the Doi-Edwards polymer model with dimension d ≥ 2 . The model was derived in the late 1970s to describe the dynamics of polymers in melts. The system contains a Navier-Stokes equation with an additional stress tensor which depend on the deformation gradient tensor and the memory function. The deformation gradient tensor satisfies a transport equation and the memory function satisfies a degenerate parabolic equation. We first proved the local well-posedness for the Doi-Edwards polymer model in Besov spaces by using the Littlewood-Paley theory. Moreover, if the initial velocity and the initial memory is small enough, we obtain a global existence result.

Objective: Catheters and wires are used extensively in cardiac catheterization procedures. Detecting their positions in fluoroscopic X-ray images is important for several clinical applications such as motion compensation and co-registration between 2D and 3D imaging modalities. Detecting the complete length of a catheter or wire object as well as electrode positions on the catheter or wire is a challenging task. Method: In this paper, an automatic detection framework for catheters and wires is developed. It is based on path reconstruction from image tensors, which are eigen direction vectors generated from a multiscale vessel enhancement filter. A catheter or a wire object is detected as the smooth path along those eigen direction vectors. Furthermore, a real-time tracking method based on a template generated from the detection method was developed. Results: The proposed framework was tested on a total of 7,754 X-ray images. Detection errors for catheters and guidewires are 0.56 0.28 mm and 0.68 0.33 mm, respectively. The proposed framework was also tested and validated in two clinical applications. For motion compensation using catheter tracking, the 2D target registration errors (TRE) of 1.8 mm 0.9 mm was achieved. For co-registration between 2D X-ray images and 3D models from MRI images, a TRE of 2.3 0.9 mm was achieved. Conclusion: A novel and fully automatic detection framework and its clinical applications are developed. Significance: The proposed framework can be applied to improve the accuracy of image-guidance systems for cardiac catheterization procedures.

Driven by the demands of deep learning, many hardware accelerators, including GPUs, have begun to include specialized tensor processing units to accelerate matrix operations. However, general-purpose GPU applications that have little or no large dense matrix operations cannot benefit from these tensor units. This article proposes Tensorox, a framework that exploits the half-precision tensor cores available on recent GPUs for approximable, non deep learning applications. In essence, a shallow neural network is trained based on the input-output mapping of the function to be approximated. The key innovation in our implementation is the use of the small and dimension-restricted tensor operations in Nvidia GPUs to run multiple instances of the approximation neural network in parallel. With the proper scaling and training methods, our approximation yielded an overall accuracy that is higher than naively running the original programs with half-precision. Furthermore, Tensorox allows for the runtime adjustment of the degree of approximation. For the 10 benchmarks we tested, we achieved speedups from 2× to 112× compared to the original in single precision floating point, while maintaining the error caused by the approximation to below 10 percent in most applications.

With the rapid advancement of spectrometers, the imaging range of the electromagnetic spectrum starts growing narrower. The reduction of electromagnetic wave energy received in a single wavelength range leads more complex noise into the generated hyperspectral image (HSI), thus causing a severe cripple in the accuracy of subsequent applications. The requirement for the HSI mixed denoising algorithm's accuracy is further lifted. To address this challenge, in this letter, we propose a novel difference continuity-regularized nonlocal tensor subspace low-rank learning (named DNTSLR) method for HSI mixed denoising. Technically, the original high-dimensional HSI data was first projected into a low-dimensional subspace spanned by a spectral difference continuous basis instead of an orthogonal basis, so the data continuity of the restored HSI spectrum and tensor low-rankness was guaranteed. Then, a cube matching strategy was employed to stack the nonlocal tensor patches from the projected coefficient tensor, and a shrinkage algorithm was used to approximate the low-rank coefficient tensor. Eventually, the subspace low-rank learning algorithm was designed to alternately separate the noise tensor and restore the latent clean low-rank HSI tensor. Extensive experiments on multiple open datasets validate that the proposed method realizes the state-of-the-art denoising accuracy for HSI.

Millimeter Wave (mmWave) massive multiple-input multiple-output (MIMO) systems with hybrid analog-digital architectures can greatly increase system capacity and communicate with multiple users at the same time. Accurate channel estimation is crucial for multi-user communications, but its accuracy is limited as the number of antennas and users increases. In the matrix high-dimensional operation for multi-user channel estimation, not only is it computationally intensive, but also the estimation accuracy is low. It makes our work turn to channel estimation of the user group within a certain region to improve the accuracy of estimation. In this paper, we propose a tensor dictionary manifold learning method for channel estimation and interference elimination of the multi-user mmWave massive MIMO system. A multi-user digital-analog mixed received signal model is presented. The tensor dictionary manifold learning scheme is proposed to model the received signal as a third-order low-rank tensor to handle the high-dimensional user, antenna, and channel. After segmentation, clustering and manifold learning, multiple tensor dictionary manifold models containing a group of user signals are fitted. Tensor dictionary manifold learning can take advantage of the inherent multi-domain properties of signals in the frequency, time, code and spatial domains to maintain inter-user correlation within a user group while reducing the high-dimensional channels of the user group. Using the convex relaxation property of the tensor alternating direction method, we propose a strategy to eliminate interference from other groups. And with the help of the multi-signal classification method, the channel parameters of user groups are obtained to improve the accuracy of multi-user channel estimation. This method can perform channel estimation for multiple users with only a few pilots, and improve the performance of the system. Numerical results confirm the good performance of this method.

Graph and subspace clustering methods have become the mainstream of multi-view clustering due to their promising performance. However, (1) since graph clustering methods learn graphs directly from the raw data, when the raw data is distorted by noise and outliers, their performance may seriously decrease; (2) subspace clustering methods use a “two-step” strategy to learn the representation and affinity matrix independently, and thus may fail to explore their high correlation. To address these issues, we propose a novel multi-view clustering method via learning a Low-Rank Tensor Graph (LRTG). Different from subspace clustering methods, LRTG simultaneously learns the representation and affinity matrix in a single step to preserve their correlation. We apply Tucker decomposition and l2;1-norm to the LRTG model to alleviate noise and outliers for learning a “clean” representation. LRTG then learns the affinity matrix from this “clean” representation. Additionally, an adaptive neighbor scheme is proposed to find the K largest entries of the affinity matrix to form a flexible graph for clustering. An effective optimization algorithm is designed to solve the LRTG model based on the alternating direction method of multipliers. Extensive experiments on different clustering tasks demonstrate the effectiveness and superiority of LRTG over seventeen state-of-the-art clustering methods.

Private individual information are increasingly exposed through high-dimensional and high-order data, with the wide deployment of learning techniques. These data are typically expressed in form of tensors, but there is no principled way to guarantee privacy for tensor-valued queries. Conventional differential privacy is typically applied to scalar values without a precise definition on the shape of the queried data. Realizing that the conventional mechanisms do not take the data structural information into account, we propose Tensor Variate Gaussian (TVG), a new (,δ)-differential privacy mechanism for tensor-valued queries. We further introduce two mechanisms based on TVG with an improved utility by imposing the unimodal differentially-private noise. With the utility space available, the proposed mechanisms can be instantiated with an optimized utility, and the optimization problem has a closed-form solution scalable to large-scale problems. Finally, we experimentally test our mechanisms on a variety of datasets and models, demonstrating that TVG is superior than other state-of-the-art mechanisms on tensor-valued queries.

Tensor factorization methods have recently gained increased popularity. A key feature that renders tensors attractive is the ability to directly model multi-relational data. In this work, we propose ParaSketch, a parallel tensor factorization algorithm that enables massive parallelism, to deal with large tensors. The idea is to compress the large tensor into multiple small tensors, decompose each small tensor in parallel, and combine the results to reconstruct the desired latent factors. Prior art in this di- rection entails potentially very high complexity in the (Gaussian) compression and final combining stages. Adopting sketching matrices for compression, the proposed method enjoys a dramatic reduction in compression complexity, and features a much lighter combining step. Moreover, theoretical analysis shows that the compressed tensors inherit latent identifiability under mild conditions, hence establishing correctness of the overall approach. Numerical experiments corroborate the theory and demonstrate the effectiveness of the proposed algorithm.

A new approximation algorithm is developed to compute the electromagnetic (EM) tensor Green's function in the biaxial anisotropic media based on the Fourier series expansion. First, a rectangular region is chosen as the computation region, in which the EM field can be expressed as a 2-D Fourier series. The Fourier coefficients can be regarded as the EM field in discrete wavenumber domain. We further obtain the EM solution in the spatial domain in the form of Fourier series. Then, we use the finite terms of the Fourier series to approximate the EM field for enhancement of computation efficiency. Because the new method avoids the numerical integration in the infinite wavenumber domain, it is more convenient to implement than other methods based on integral transform. Finally, the spatial distribution of the tensor Green's function for a transversely isotropic medium is presented to verify the proposed approach. The agreements between the results obtained by the present method and the analytic solution demonstrate the validity and the robustness of our algorithm.

Due to the unequispaced pulse repetition interval (PRI), the low-oversampling ratio and the range-variant blockage, the echo of the low-oversampled staggered SAR (LS-SAR) is nonuniformly sampled with sub-Nyquist and range-variant rate. However, the existing LS-SAR processing methods lack robustness with regards to the scenario type and the PRI variation mode. In this article, a compressive-sensing-based image reconstruction method for the LS-SAR is proposed. First, a hybrid-domain model (HDM) of the LS-SAR echo is presented. In the HDM, the coupled range cell migration (RCM), the unequispaced PRI, and the conflict blockage are formulated as the matrix multiplications with a 3-D tensor, a 2-D matrix, and a Hadamard product, respectively. Based on the HDM, the image reconstruction is realized through the 2-D fast iterative shrinkage thresholding algorithm (ISTA), in which the gradient is derived by exploiting the properties of the tensor and matrix trace. The fast Fourier transform (FFT) and the nonuniform FFT are implemented to accelerate the computation. Due to good accommodation of the RCM and the LS-SAR sampling characteristics, the proposed method can work well for various PRI variation modes and scenario types. Simulations using the point scatter and the distributed target with wide-swath extension demonstrate the effectiveness as well as the robustness of the proposed method.

Most of the algorithms for hyperspectral anomaly detection (AD) are based on the original spectral signatures which may suffer noise contamination. In recent years, some AD algorithms based on deep learning (DL) and tensor have achieved satisfactory results. In this letter, an algorithm using fractional Fourier transform (FrFT) and transferred convolutional neural network based on tensor (FrFTTCNNT) is proposed for hyperspectral AD. First, the test block of each test point in hyperspectral imagery (HSI) is transformed into 1-D vector and a higher dimensional data set with more spatial information is obtained. Furthermore, the higher dimensional data set is dimensionally reduced by principal component analysis (PCA) to remove the redundancy of spectral bands. Then, the lower dimensional data set after PCA is transformed by FrFT and the nonstationary noise in the fractional Fourier domain (FrFD) can be better suppressed which may increase the discrimination between background and targets. Finally, in the FrFD, transferred CNN based on tensor (TCNNT) is employed for the final results. Experiments conducted on three hyperspectral data sets show the superiority of the proposed FrFTTCNNT.

Tradeoffs between the spatial and temporal resolutions of current satellite instruments limit our ability to conduct high-quality and continuous monitoring of the earth's surface dynamics. Spatiotemporal image fusion has become increasingly necessary to obtain remote sensing images with high spatiotemporal resolution. However, current learning-based methods concentrate on predicting images only from spatial similarity and neglect spectral correlations of remote sensing images, leading to significant spectral information loss. In this article, we develop a novel nonlocal tensor sparse representation-based semicoupled dictionary learning approach (SCDNTSR) for spatiotemporal fusion. In the SCDNTSR method, the spectral correlation and the spatial similarity of the nonlocal similar cubes are simultaneously exploited through the tensor-tensor product-based tensor sparse representation. Furthermore, the semicoupled mapping prior knowledge of sparse coefficients across the high- and low-spatial resolution (HSR\backslashLSR) image spaces is exploited with the coupled dictionary to constrain the similarity of sparse coefficients to improve the prediction performance. In addition, to capture additional prior spatial information, the SCDNTSR provides a new method to determine the degradation relationship between the target HSR and LSR difference images with the help of the known HSR and LSR difference images. The proposed SCDNTSR method was tested on real datasets at both the Coleambally Irrigation Area study site and the Lower Gwydir Catchment study site. Results show that the proposed method outperforms five state-of-the-art methods, especially in maintaining the spectral information, proving the feasibility of integrating the degradation relationship, spatio-spectral-nonlocal correlation, and semicoupled mapping priors of the multisource data into the proposed model.

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.

This article proposed a nonconvex variational model for pansharpening with spatial and spectral gradient difference-induced nonconvex sparsity priors (PSSGDNSP), which can fuse the panchromatic (Pan) and low-resolution (LR) multispectral (MS) images to generate the high-resolution (HR) MS image. More particularly, the proposed PSSGDNSP model exploits the spatial gradient difference-induced nonconvex l1/2 sparsity prior between HR MS and Pan, and the spectral gradient difference-induced nonconvex l1/2 sparsity prior between HR and LR MS. Consequently, our proposed PSSGDNSP model well preserves both the spatial and spectral information. In fact, our proposed band-coupled model treats the MS image like a third-order tensor so that the intrinsic band correlation of the MS image can be fully kept. Moreover, we solve our proposed PSSGDNSP model by applying the alternating direction method of multipliers (ADMM) method. Finally, the experiments fully validate the superiority and performance of our proposed PSSGDNSP method.

Non-negative Tucker decomposition (NTD) is one of the most popular techniques for tensor data representation. To enhance the representation ability of NTD by multiple intrinsic cues, that is, manifold structure and supervisory information, in this article, we propose a generalized graph regularized NTD (GNTD) framework for tensor data representation. We first develop the unsupervised GNTD (UGNTD) method by constructing the nearest neighbor graph to maintain the intrinsic manifold structure of tensor data. Then, when limited must-link and cannot-link constraints are given, unlike most existing semisupervised learning methods that only use the pregiven supervisory information, we propagate the constraints through the entire dataset and then build a semisupervised graph weight matrix by which we can formulate the semisupervised GNTD (SGNTD). Moreover, we develop a fast and efficient alternating proximal gradient-based algorithm to solve the optimization problem and show its convergence and correctness. The experimental results on unsupervised and semisupervised clustering tasks using four image datasets demonstrate the effectiveness and high efficiency of the proposed methods.

Hyperspectral images (HSIs) are often contaminated by several types of noise, which significantly limits the accuracy of subsequent applications. Recently, low-rank modeling based on tensor singular value decomposition (T-SVD) has achieved great success in HSI restoration. Most of them use the convex and nonconvex surrogates of the tensor rank, which cannot well approximate the tensor singular values and obtain suboptimal restored results. We suggest a novel HSI restoration model by introducing a fibered rank constrained tensor restoration framework with an embedded plug-and-play (PnP)-based regularization (FRCTR-PnP). More precisely, instead of using the convex and nonconvex surrogates to approximate the fibered rank, the proposed model directly constrains the tensor fibered rank of the solution, leading to a better approximation to the original image. Since exploiting the low-fibered-rankness of HSI is mainly to capture the global structure, we further employ an implicit PnP-based regularization to preserve the image details. Particularly, the above two building blocks are complementary to each other, rather than isolated and uncorrelated. Based on the alternating direction multiplier method (ADMM), we propose an efficient algorithm to tackle the proposed model. For robustness, we develop a three-directional randomized T-SVD (3DRT-SVD), which preserves the intrinsic structure of the clean HSI and removes partial noise by projecting the HSI onto a low-dimensional essential subspace. Extensive experimental results including simulated and real data demonstrate that the proposed method achieves superior performance over compared methods in terms of quantitative evaluation and visual inspection.

Zinc phosphide (Zn3P2) is an II-V compound semiconductor with promising photovoltaic and thermoelectric applications. Its complex structure is susceptible to facile defect formation, which plays a key role in further optimization of the material. Raman spectroscopy can be effectively used for defect characterization. However, the Raman tensor of Zn3P2, which determines the intensity of Raman peaks and anisotropy of inelastic light scattering, is still unknown. In this paper, we use angle-resolved polarization Raman measurements on stoichiometric monocrystalline Zn3P2 thin films and first-principle calculations to obtain Raman tensor of Zn3P2. This has allowed determination of the Raman tensor elements characteristic for the A1g, B1g and B2g vibrational modes. These results have been compared with the theoretically obtained Raman tensor elements and simulated Raman spectra from the lattice-dynamics calculations using first-principles force constants. Excellent agreement is found between the experimental and simulated Raman spectra of Zn3P2 for various polarization configurations, providing a platform for future characterization of the defects in this material.

Existing infrared small-target detection methods tend to perform unsatisfactorily when encountering complex scenes, mainly due to the following: 1) the infrared image itself has a low signal-to-noise ratio (SNR) and insufficient detailed/texture knowledge; 2) spatial and structural information is not fully excavated. To avoid these difficulties, an effective method based on three-order tensor creation and Tucker decomposition (TCTD) is proposed, which detects targets with various brightness, spatial sizes, and intensities. In the proposed TCTD, multiple morphological profiles, i.e., diverse attributes and different shapes of trees, are designed to create three-order tensors, which can exploit more spatial and structural information to make up for lacking detailed/texture knowledge. Then, Tucker decomposition is employed, which is capable of estimating and eliminating the major principal components (i.e., most of the background) from three dimensions. Thus, targets can be preserved on the remaining minor principal components. Image contrast is further enhanced by fusing the detection maps of multiple morphological profiles and several groups with discontinuous pruning values. Extensive experiments validated on two synthetic data and six real data sets demonstrate the effectiveness and robustness of the proposed TCTD.

This letter proposed a spatial and spectral anisotropic tensor total variation (SSATTV) driven adaptive pansharpening model for the fusion of low-resolution (LR) multispectral (MS) and panchromatic (Pan) images to the high-resolution (HR) MS images. Except for the local spectral consistency constraint-based fidelity term between HR and LR MS used for spectral preservation, the proposed model reformulated the adaptive linear constraint-based fidelity term between Pan and HR MS with the tensor-representation modeling, and particularly proposed a SSATTV prior term which imposed the spatial anisotropic TV prior between HR MS and Pan for spatial gradient feature preservation, and the spectral TV sparsity prior on HR MS for further spectral feature preservation. Furthermore, the proposed model was efficiently solved via the alternating direction method of multipliers (ADMM) scheme. Specifically, the experiments validated the superiority of proposed SSATTV method.

The goal of this paper is to investigate a new model, called generalized polynomial complementarity problems over a polyhedral cone and denoted by GPCPs, which is a natural extension of the polynomial complementarity problems and generalized tensor complementarity problems. Firstly, the properties of the set of all $$R^{K}_{{\varvec{0}}}$$ -tensors are investigated. Then, the nonemptiness and compactness of the solution set of GPCPs are proved, when the involved tensor in the leading term of the polynomial is an $$ER^{K}$$ -tensor. Subsequently, under fairly mild assumptions, lower bounds of solution set via an equivalent form are obtained. Finally, a local error bound of the considered problem is derived. The results presented in this paper generalize and improve the corresponding those in the recent literature.

In this paper, a tensor flow-based fast reroute model with multimedia quality protection is proposed. In the model, the conditions for implementing a multipath routing strategy and flow conservation are introduced taking into account possible packet loss at the network nodes and preventing overloading communication links both when using the primary and backup routes. At the same time, the novelty of the proposed solution is the formalization of the conditions of protection of the Quality of Experience level in terms of multimedia quality along the primary and backup routes. These conditions have been obtained during the tensor formalization of the network, which made it possible to calculate the quality of service indicators: packet loss probabilities, as well as the average end-to-end delay for audio and video flows transmitted in the multimedia session using the primary and backup routes, respectively. As a criterion for the optimality of the obtained solutions, a condition has been selected related to maximizing the overall performance of the infocommunication network. The results of the research of the proposed model confirm the adequacy of the numerical research results obtained for solving the problem of fast rerouting with link/node protection.

Big data has been continuously generated from the rapidly developing of cloud/fog/edge computing, Internet of Things (IoT) and 5G technology. This big data not only brings great benefits and opportunities to human beings but also brings many risks and challenges. One major challenge is how to represent and treat higher-order and heterogeneous data from multi-sources. Tensors are emerging as powerful tools for representation and modeling of this data. Tensor decomposition can be used to extract potentially useful information from this data. Thus, it has attracted much attention from the big data community. The main target of this paper is to propose a novel data fusion framework to solve several main challenges of CPSS data applications, including CPSS big data representation, fusion, efficient computing, storage, robustness, and security issues. In this paper, we use many graphics to visualize complex tensor decomposition and transformation processes. The visualization may help the readers better understand tensor and tensor decomposition. It also provides a general guideline and a good starting point for those who are interested in tensor and tensor decomposition. Specifically, we first introduce the most extensively used matrix and tensor decomposition methods. Second, the current popular data fusion methods are reviewed and summarized. Third, we propose a novel tensor-network-conversion-based data fusion approach which can simultaneously analyze multiple matrices and multiple tensors. To better understand this approach, we give a brief review of tensor network. Fourth, based on the proposed approach, a novel CPSS big data fusion framework is proposed in this paper. Meanwhile, we verify it by a concise case study. Finally, some challenges and open problems of the proposed framework are discussed. The discussion also includes some exciting future research directions in the big data fusion field.