Tensor Framework Research Articles

Alteration detection of tensor dependence structure via sparsity-exploited reranking algorithm

Tensor-valued data arise frequently from a wide variety of scientific applications, and many among them can be translated into an alteration detection problem of tensor dependence structures. In this article, we formulate the problem under the popularly adopted tensor-normal distributions and aim at two-sample correlation/partial correlation comparisons of tensor-valued observations. Through decorrelation and centralization, a separable covariance structure is employed to pool sample information from different tensor modes to enhance the power of the test. Additionally, we propose a novel Sparsity-Exploited Reranking Algorithm (SERA) to further improve the multiple testing efficiency. Such efficiency gain is achieved by incorporating a carefully constructed auxiliary tensor sequence to rerank the p-values. Besides the tensor framework, SERA is also generally applicable to a wide range of two-sample large-scale inference problems with sparsity structures, and is of independent interest. The asymptotic properties of the proposed test are derived and the algorithm is shown to control the false discovery at the pre-specified level. We demonstrate the efficacy of the proposed method through intensive simulations and two scientific applications.

CP decomposition-based algorithms for completion problem of motion capture data

Motion capture (MoCap) technology is an essential tool for recording and analyzing movements of objects or humans. However, MoCap systems frequently encounter the challenge of missing data, stemming from mismatched markers, occlusion, or equipment limitations. Recovery of these missing data is imperative to maintain the reliability and integrity of MoCap recordings. This paper introduces a novel application of the tensor framework for MoCap data completion. We propose three completion algorithms based on the canonical polyadic (CP) decomposition of tensors. The first algorithm utilizes CP decomposition to capture the low-rank structure of the tensor. However, relying only on low-rank assumptions may be insufficient to deal with complex motion data. Thus, we propose two modified CP decompositions that incorporate additional information, SmoothCP and SparseCP decompositions. SmoothCP integrates piecewise smoothness prior, while SparseCP incorporates sparsity prior, each aiming to improve the accuracy and robustness of MoCap data recovery. To compare and evaluate the merit of the proposed algorithms over other tensor completion methods in terms of several evaluation metrics, we conduct numerical experiments with different MoCap sequences from the CMU motion capture dataset.

FuBay: An Integrated Fusion Framework for Hyperspectral Super-Resolution Based on Bayesian Tensor Ring.

Fusion with corresponding finer-resolution images has been a promising way to enhance hyperspectral images (HSIs) spatially. Recently, low-rank tensor-based methods have shown advantages compared with other kind of ones. However, these current methods either relent to blind manual selection of latent tensor rank, whereas the prior knowledge about tensor rank is surprisingly limited, or resort to regularization to make the role of low rankness without exploration on the underlying low-dimensional factors, both of which are leaving the computational burden of parameter tuning. To address that, a novel Bayesian sparse learning-based tensor ring (TR) fusion model is proposed, named as FuBay. Through specifying hierarchical sprasity-inducing prior distribution, the proposed method becomes the first fully Bayesian probabilistic tensor framework for hyperspectral fusion. With the relationship between component sparseness and the corresponding hyperprior parameter being well studied, a component pruning part is established to asymptotically approaching true latent rank. Furthermore, a variational inference (VI)-based algorithm is derived to learn the posterior of TR factors, circumventing nonconvex optimization that bothers the most tensor decomposition-based fusion methods. As a Bayesian learning methods, our model is characterized to be parameter tuning-free. Finally, extensive experiments demonstrate its superior performance when compared with state-of-the-art methods.

Fractional Hermite–Hadamard, Newton–Milne, and Convexity Involving Arithmetic–Geometric Mean-Type Inequalities in Hilbert and Mixed-Norm Morrey Spaces ℓq(·)(Mp(·),v(·)) with Variable Exponents

Function spaces play a crucial role in the study and application of mathematical inequalities. They provide a structured framework within which inequalities can be formulated, analyzed, and applied. They allow for the extension of inequalities from finite-dimensional spaces to infinite-dimensional contexts, which is crucial in mathematical analysis. In this note, we develop various new bounds and refinements of different well-known inequalities involving Hilbert spaces in a tensor framework as well as mixed Moore norm spaces with variable exponents. The article begins with Newton–Milne-type inequalities for differentiable convex mappings. Our next step is to take advantage of convexity involving arithmetic–geometric means and build various new bounds by utilizing self-adjoint operators of Hilbert spaces in tensorial frameworks for different types of generalized convex mappings. To obtain all these results, we use Riemann–Liouville fractional integrals and develop several new identities for these operator inequalities. Furthermore, we present some examples and consequences for transcendental functions. Moreover, we developed the Hermite–Hadamard inequality in a new and significant way by using mixed-norm Moore spaces with variable exponent functions that have not been developed previously with any other type of function space apart from classical Lebesgue space. Mathematical inequalities supporting tensor Hilbert spaces are rarely examined in the literature, so we believe that this work opens up a whole new avenue in mathematical inequality theory.

Tucker tensor decomposition with rank estimation for sparse hyperspectral unmixing

ABSTRACT Hyperspectral image unmixing gives increasing focus on the structural integrity of the spatial-spectral information. A large number of methods exploit the sparse and low-rank properties of the abundance vectors, abundance matrices and abundance maps, which are first-order or second-order structures, and are hard to preserve the integrated structure of the abundance. The abundance tensor formed by overlapping the abundance maps of endmembers sequentially is a more appropriate expression of the abundance. However, hyperspectral unmixing studies based on the abundance tensor are still in the minority, and most of these methods either inherently degrade the tensor structure to some extent, which is hard to distinguish the low-rankness of different dimensions; or are based on tensor decomposition methods, which always require a pre-estimation of the tensor rank, thus the unmixing accuracy depends on the rank estimation. In this article, we propose an unmixing model on tensor framework, in which the abundance tensor is approximated by a truncated Tucker tensor decomposition (TTD). Since the generated factor matrices are relatively low-rank, the low-rankness exploration of the abundance tensor is achieved potentially by this way. Moreover, we further improve the unmixing performance by a suitable rank estimation. The more flexible algorithm Tucker tensor decomposition with rank estimation for sparse hyperspectral unmixing (TTDRE) is proposed to differentiate the low-rankness of different dimensions of the abundance tensor while preserving the structural integrity. Experiments on synthetic and real data illustrate the effectiveness of our method.

Kalman Filter Using a Third-Order Tensorial Decomposition of the Impulse Response

For system identification problems associated with long-length impulse responses, the recently developed decomposition-based technique that relies on a third-order tensor (TOT) framework represents a reliable choice. It is based on a combination of three shorter filters, which merge their estimates in tandem with the Kronecker product. In this way, the global impulse response is modeled in a more efficient manner, with a significantly reduced parameter space (i.e., fewer coefficients). In this paper, we further develop a Kalman filter based on the TOT decomposition method. As compared to the recently designed recursive least-squares (RLS) counterpart, the proposed Kalman filter achieves superior performance in terms of the main criteria (e.g., tracking and accuracy). In addition, it significantly outperforms the conventional Kalman filter, while also having a lower computational complexity. Simulation results obtained in the context of echo cancellation support the theoretical framework and the related advantages.

A microdamage model for FCC single crystals considering a mixed failure mechanism of slip and cleavage

In this study, a microdamage model is newly proposed to predict the failure process of FCC single crystals. Two micro damage mechanisms, slip and cleavage, are both involved in the model. A microscopic damage variable is first defined at the slip plane scale, and a damage coupled crystal plasticity constitutive equation in a co-rotational tensor framework is then established. A damage driving force consisting of generalized energies related to the resolved shear stress and the positive resolved normal stress is proposed to reflect the mixed failure mechanism of FCC single crystals induced by slip and cleavage. The damage evolution equation is then derived in the framework of thermodynamics. An explicit algorithm for the damage coupled crystal plasticity finite element method is given. The proposed microdamage model is utilized to predict the slip-dominant fracture behavior of a single crystal copper and the cleavage-dominant fracture behavior of a nickel-based single crystal alloy. The damage evolution process of slip planes is given and the influence mechanism is revealed. The calculated results agree well with the experimental data, which verifies the good capability of the proposed model in describing and predicting the multi failure processes of FCC metals. Finally, a 2D fracture simulation of a single crystal notched specimen under shear loading is conducted, and the effect of damage parameters on the fracture modes is also discussed.

Non-electrocardiogram-gated, free-breathing, off-resonance reduced, high-resolution, whole-heart myocardial T2 * mapping at 3 T within 5 min.

Widely used conventional 2D T2 * approaches that are based on breath-held, electrocardiogram (ECG)-gated, multi-gradient-echo sequences are prone to motion artifacts in the presence of incomplete breath holding or arrhythmias, which is common in cardiac patients. To address these limitations, a 3D, non-ECG-gated, free-breathing T2 * technique that enables rapid whole-heart coverage was developed and validated. A continuous random Gaussian 3D k-space sampling was implemented using a low-rank tensor framework for motion-resolved 3D T2 * imaging. This approach was tested in healthy human volunteers and in swine before and after intravenous administration of ferumoxytol. Spatial-resolution matched T2 * images were acquired with 2-3-fold reduction in scan time using the proposed T2 * mapping approach relative to conventional T2 * mapping. Compared with the conventional approach, T2 * images acquired with the proposed method demonstrated reduced off-resonance and flow artifacts, leading to higher image quality and lower coefficient of variation in T2 *-weighted images of the myocardium of swine and humans. Mean myocardial T2 * values determined using the proposed and conventional approaches were highly correlated and showed minimal bias. The proposed non-ECG-gated, free-breathing, 3D T2 * imaging approach can be performed within 5 min or less. It can overcome critical image artifacts from undesirable cardiac and respiratory motion and bulk off-resonance shifts at the heart-lung interface. The proposed approach is expected to facilitate faster and improved cardiac T2 * mapping in those with limited breath-holding capacity or arrhythmias.

Fusion-based tensor radiomics using reproducible features: Application to survival prediction in head and neck cancer

BackgroundNumerous features are commonly generated in radiomics applications as applied to medical imaging, and identification of robust radiomics features (RFs) can be an important step to derivation of reliable, reproducible solutions. In this work, we utilize a tensor radiomics (TR) framework, where numerous fusions are explored, to generate different flavours of RFs, and we aimed to identify RFs that are robust to fusion techniques in head and neck cancer. Overall, we aimed to predict progression-free survival (PFS) using Hybrid Machine Learning Systems (HMLS) and reproducible RFs. MethodsThe study was performed on 408 patients with head and neck cancer from The Cancer Imaging Archive. After image preprocessing, 15 fusion techniques were employed to combine Positron Emission Tomography (PET) and Computed Tomography (CT) images. Subsequently, 215 RFs were extracted through a standardized radiomics software, with 17 ‘flavours’ generated using PET-only, CT-only, and 15 fused PET&CT images. The variability of RFs across flavours was studied using the Intraclass Correlation Coefficient (ICC). Furthermore, the features were categorized into seven reliability groups, 106 reproducible RFs with ICC>0.75 were selected, highly correlated flavours were removed, Principal Component Analysis was used to convert 17 flavours to 1 attribute, the polynomial function was utilized to increase RFs, and Analysis of variance (ANOVA) was used to select the relevant attributes. Finally, 3 classifiers including Random Forest (RFC), Logistic regression (LR), and Multi-layer perceptron were applied to the preselected relevant attributes to predict binary PFS. In 5-fold cross-validation, 80% of 4 divisions were utilized to train the model, and the remaining 20% was utilized to evaluate the model. Further, the remaining fold was used for external nested testing. ResultsReliability analysis indicated that most morphological features belong to the high-reliability category. By contrast, local intensity and statistical features extracted from images belong to the low-reliability category. In the tensor framework, the highest 5-fold cross-validation accuracy of 76.7%±3.3% with an external nested testing of 70.6%±6.7% resulted from the reproducible TR+polynomial function+ANOVA+LR algorithm while the accuracy of 70.0%±4.2% with the external nested testing of 67.7%±4.9% was achieved through the PCA fusion+RFC (non-tensor paradigm). ConclusionsThis study demonstrated that using reproducible RFs as utilized within a tensor fusion radiomics framework, linked with ANOVA and LR, added value to prediction of progression-free survival outcome in head and neck cancer patients.

Nonconvex Tensor Relative Total Variation for Image Completion

Image completion, which falls to a special type of inverse problems, is an important but challenging task. The difficulties lie in that (i) the datasets usually appear to be multi-dimensional; (ii) the unavailable or corrupted data entries are randomly distributed. Recently, low-rank priors have gained importance in matrix completion problems and signal separation; however, due to the complexity of multi-dimensional data, using a low-rank prior by itself is often insufficient to achieve desirable completion, which requires a more comprehensive approach. In this paper, different from current available approaches, we develop a new approach, called relative total variation (TRTV), under the tensor framework, to effectively integrate the local and global image information for data processing. Based on our proposed framework, a completion model embedded with TRTV and tensor p-shrinkage nuclear norm minimization with suitable regularization is established. An alternating direction method of multiplier (ADMM)-based algorithm under our framework is presented. Extensive experiments in terms of denoising and completion tasks demonstrate our proposed method are not only effective but also superior to existing approaches in the literature.

Change-Point Detection for Multi-Way Tensor-Based Frameworks.

Graph-based change-point detection methods are often applied due to their advantages for using high-dimensional data. Most applications focus on extracting effective information of objects while ignoring their main features. However, in some applications, one may be interested in detecting objects with different features, such as color. Therefore, we propose a general graph-based change-point detection method under the multi-way tensor framework, aimed at detecting objects with different features that change in the distribution of one or more slices. Furthermore, considering that recorded tensor sequences may be vulnerable to natural disturbances, such as lighting in images or videos, we propose an improved method incorporating histogram equalization techniques to improve detection efficiency. Finally, through simulations and real data analysis, we show that the proposed methods achieve higher efficiency in detecting change-points.

Tensorial Multiview Representation for Saliency Detection via Nonconvex Approach.

In the study of salient object detection, multiview features play an important role in identifying various underlying salient objects. As to current common patch-based methods, all different features are handled directly by stacking them into a high-dimensional vector to represent related image patches. These approaches ignore the correlations inhering in the original spatial structure, which may lead to the loss of certain underlying characterization such as view interaction. In this article, different from currently available approaches, a tensorial feature representation framework is developed for the salient object detection in order to better explore the complementary information of multiview features. Under the tensor framework, a tensor low-rank constraint is applied to the background to capture its intrinsic structure, a tensor group sparsity regularization is posed on the salient part, and a tensorial sliced Laplacian regularization is then introduced to enlarge the gap between the subspaces of the background and salient object. Moreover, a nonconvex tensor Log-determinant function, instead of the tensor nuclear norm, is adopted to approximate the tensor rank for effectively suppressing the confusing information resulted from underlying complex backgrounds. Further, we have deduced the closed-form solution of this nonconvex minimization problem and established a feasible algorithm whose convergence is mathematically proven. Experiments on five well-known public datasets are provided and the simulations demonstrate that our method outperforms the latest unsupervised handcrafted features-based methods in the literature. Furthermore, our model is flexible with various deep features and is competitive with the state-of-the-art approaches.

Guaranteed Functional Tensor Singular Value Decomposition

This article introduces the functional tensor singular value decomposition (FTSVD), a novel dimension reduction framework for tensors with one functional mode and several tabular modes. The problem is motivated by high-order longitudinal data analysis. Our model assumes the observed data to be a random realization of an approximate CP low-rank functional tensor measured on a discrete time grid. Incorporating tensor algebra and the theory of reproducing kernel Hilbert space (RKHS), we propose a novel RKHS-based constrained power iteration with spectral initialization. Our method can successfully estimate both singular vectors and functions of the low-rank structure in the observed data. With mild assumptions, we establish the non-asymptotic contractive error bounds for the proposed algorithm. The superiority of the proposed framework is demonstrated via extensive experiments on both simulated and real data. Supplementary materials for this article are available online.

Low-Rank Tensor Subspace Decomposition With Weighted Group Sparsity for the Acceleration of Non-Cartesian Dynamic MRI.

Dynamic MR imaging often requires long scan time, and acceleration of data acquisition is highly desirable in clinical applications. We proposed a Low-rank Tensor subspace decomposition with Weighted Group Sparsity (LTWGS) algorithm for non-Cartesian dynamic MRI. The proposed algorithm introduces the weighted group sparse constraints together with the subspace decomposition technique into the framework of low-rank tensor and sparse decomposition to better utilize the sparsity in the data. LTWGS increases the PSNR values by 1.97 dB, 2.03 dB, and 2.83 dB compared with PROST (patch-based reconstruction), SRTPCA (smooth robust tensor principal component analysis), and LRTES (low-rank tensor with "explicit subspace") in the dynamic abdominal imaging at an acceleration rate R = 25. LTWGS increases the PSNR values by 2.42 dB and 3.57 dB compared with PROST and LRTES in DCE liver imaging at R = 25. LTWGS increases the PSNR values by 1.40 dB and 1.96 dB compared with PROST and SRTPCA in cardiac cine imaging at R = 25. Jointly using group sparsity and sparsity can obtain better results than that using group sparsity alone, and weighted regularization can achieve better results than that without weighted regularization. The proposed algorithm results in reduced reconstruction error and improved image structural similarity in comparison with several state-of-the-art methods at relatively high acceleration factors. The proposed algorithm has the potential in various dynamic MRI application scenarios.

Dynamic Community Detection for Brain Functional Networks During Music Listening With Block Component Analysis

The human brain can be described as a complex network of functional connections between distinct regions, referred to as the brain functional network. Recent studies show that the functional network is a dynamic process and its community structure evolves with time during continuous task performance. Consequently, it is important for the understanding of the human brain to develop dynamic community detection techniques for such time-varying functional networks. Here, we propose a temporal clustering framework based on a set of network generative models and surprisingly it can be linked to Block Component Analysis to detect and track the latent community structure in dynamic functional networks. Specifically, the temporal dynamic networks are represented within a unified three-way tensor framework for simultaneously capturing multiple types of relationships between a set of entities. The multi-linear rank- (Lr, Lr, 1) block term decomposition (BTD) is adopted to fit the network generative model to directly recover underlying community structures with the specific evolution of time from the temporal networks. We apply the proposed method to the study of the reorganization of the dynamic brain networks from electroencephalography (EEG) data recorded during free music listening. We derive several network structures ( Lr communities in each component) with specific temporal patterns (described by BTD components) significantly modulated by musical features, involving subnetworks of frontoparietal, default mode, and sensory-motor networks. The results show that the brain functional network structures are dynamically reorganized and the derived community structures are temporally modulated by the music features. The proposed generative modeling approach can be an effective tool for describing community structures in brain networks that go beyond static methods and detecting the dynamic reconfiguration of modular connectivity elicited by continuously naturalistic tasks.

Ever-reaching probabilities and mean first passage times of higher order ergodic Markov chains

Within a tensor framework, the notions of ergodicity, ever-reaching probabilities, and mean first passage times of Markov chains have recently been extended to higher order Markov chains. In this paper, we establish the relations between ergodicity of a higher order Markov chain and its ever-reaching probabilities and mean first passage times. In particular, we demonstrate the key role ergodicity plays in the existence and uniqueness of the mean first passage times. We exploit the structure of the tensor equation governing the mean first passage times and propose a block iterative algorithm and a block direct algorithm for computing these quantities. We show that ergodicity guarantees the convergence of the block iterative algorithm. Some illustrative numerical results comparing these algorithms with a brute-force direct method are also given.

A PCA-based fuzzy tensor evaluation model for multiple-criteria group decision making

Multiple-criteria group decision-making (MCGDM) problems mainly consist of multiple factors and multiple Decision Makers (DMs) or Users, for which dimension extension is necessary when considering all the entries of DMs together. Tensor, a generalized form of a matrix, displays a multi-way array item, which is the most suitable and practical way to represent high-dimensional data without losing any information. In this paper, we first reduce the dimension through Principal Component Analysis (PCA), which helps consider the most-informative criteria. Then, we reintroduced the tensor as a fuzzy-form tensor for the MCGDM problem because of user information uncertainty. We choose Interval-valued Neutrosophic Fuzzy numbers (IVNFNs) as the basis for the tensor form because of their ability to distinguish between truth, indeterminacy, and falsity in the data. Lastly, a Generalized Interval-valued Neutrosophic Fuzzy Weighted Geometric (GIVNFWG) operator is defined. Moreover, a generalized framework for fuzzy-form tensors for high-dimensional MCGDM problems is proposed. The feasibility and efficiency of this proposed process is illustrated for a real-world MCGDM problem of ranking the most efficient Third Party Reverse Logistics Partners (3PRLPs), i.e., recycled fiber-based paper mills for the packaging industry. The obtained results are according to the experts and are validated using sensitivity analysis. This analysis facilitates in assessing the impact on the overall ranking performance of 3PRLPs by considering various combinations of environment and technological sub-criteria.

Tensor-Based Harmonic Analysis of Distribution Systems

Over the past few decades, there have been rapid advances in solid-state technology as well as a reduction in cost. This, coupled with the functionality and efficiency improvements they afford, has resulted in a massive increase in the use of electronic devices. Where traditionally, there were a few well-known nonlinear loads that needed to be considered, now there are numerous low-power devices. Although individually insignificant, collectively, they are very significant. This paper presents a tensor-based harmonic analysis approach that is capable of capturing important interactions while being computationally efficient enough to model a large distribution system. Numerical experiments are used to highlight the advantages of the tensor framework. Numerous papers have investigated the tensor parametrisation or its mathematical equivalent—harmonically coupled admittance matrices (also known as frequency coupling matrices). However, this paper, for the first time, demonstrates how these models can be applied to perform harmonic modelling of a complete low voltage (LV) distribution system.

A μ-mode BLAS approach for multidimensional tensor-structured problems

In this manuscript, we present a common tensor framework which can be used to generalize one-dimensional numerical tasks to arbitrary dimension d by means of tensor product formulas. This is useful, for example, in the context of multivariate interpolation, multidimensional function approximation using pseudospectral expansions and solution of stiff differential equations on tensor product domains. The key point to obtain an efficient-to-implement BLAS formulation consists in the suitable usage of the μ-mode product (also known as tensor-matrix product or mode-n product) and related operations, such as the Tucker operator. Their MathWorks MATLAB®/GNU Octave implementations are discussed in the paper, and collected in the package KronPACK. We present numerical results on experiments up to dimension six from different fields of numerical analysis, which show the effectiveness of the approach.

High-order tensor low-rank approximation with application in color video recovery

Video denoising is an elementary but critical task in computer vision and has been widely studied in recent years. However, the existing denoising methods have inevitable drawbacks: some need to predefine rank, some ignore the local information, and most cannot deal with higher-order data. To overcome these shortcomings, we consider two high-order tensor low-rank approximation methods, aiming to achieve color video denoising in a mixed noise environment. First, we establish a high-order tensor framework. Based on this framework, high-order tensor robust principal component analysis (HRPCA) is proposed. Although HRPCA is capable of processing high-order data, there is still a loss of recovery details. Then, we develop another method called high-order tensor low-rank approximation with total variation regularization (HTV). In particular, the TV consists of frontal total variation (FTV) and global total variation (GTV), thus extending the HTV into HFTV and HGTV, respectively. Extensive experimental results of color videos show that the HRPCA and HTV are more efficient in dealing with denoising problems than other state-of-the-art methods.