Tensor Matrix Research Articles

AutoGFI: Streamlined Generalized Fiducial Inference for Modern Inference Problems in Models with Additive Errors

The concept of fiducial inference was introduced by R. A. Fisher in the 1930s to address the perceived limitations of Bayesian inference, particularly the need for subjective prior distributions in cases with limited prior information. However, Fisher’s fiducial approach lost favor due to complications, especially in multi-parameter problems. With renewed interest in fiducial inference in the 2000s, generalized fiducial inference (GFI) emerged as a promising extension of Fisher’s ideas, offering new solutions for complex inference challenges. Despite its potential, GFI’s adoption has been hindered by demanding mathematical derivations and complex implementation requirements, such as Markov Chain Monte Carlo (MCMC) algorithms. This paper introduces AutoGFI, a streamlined variant of GFI designed to simplify its application across various inference problems with additive noise. AutoGFI ’s accessibility lies in its simplicity—requiring only a fitting routine—making it a feasible option for a wider range of researchers and practitioners. To demonstrate its efficacy, AutoGFI is applied to three challenging problems: tensor regression, matrix completion, and network cohesion regression. These case studies showcase AutoGFI ’s competitive performance against specialized solutions, highlighting its potential to broaden the application of GFI in practical domains, ultimately enriching the statistical inference toolkit.

Research on the Elastic Properties of Deep Rock Composites Under High Pressure

This paper presents a three-dimensional model of the definitive structure of gypsum, calcite and quartz crystals, which are the primary components of deep rock mass. The optimization of the primary components of deep rock mass material was performed under pressures ranging from 0 to 120 MPa using the first principles method. This led to the determination of the structural constants of the primary component crystals within the deep rock material. Furthermore, the variation rule of mechanical parameters, including elastic constants, for the primary components of deep-seated rock materials under different pressures was also investigated. The findings indicate a clear downward trend in the partial crystal structure parameters of deep-seated rock materials with increasing pressure. The primary components of each rock mass material display disparate compressibility properties in distinct directions when subjected to uniaxial compression. In conclusion, the anisotropic elastic properties of gypsum crystals, calcite crystals and quartz crystals under different pressures were investigated. The anisotropic cloud diagrams of bulk modulus, elastic modulus and shear modulus were presented for the three types of rock material at 0 MPa, 60 MPa and 120 MPa, respectively. In conclusion, we derived the theoretical equations for the elastic tensor matrix of composite materials using the principle of superposition. Furthermore, we studied and obtained the elastic tensor matrix for composite materials within the deep rock mass. These results provide essential theoretical guidance for the development of underground space and the analysis of deep rock dynamics.

Dual-Grained Lightweight Strategy.

Removing redundant parameters and computations before the model training has attracted a great interest as it can effectively reduce the storage space of the model, speed up the training and inference of the model, and save energy consumption during the running of the model. In addition, the simplification of deep neural network models can enable high-performance network models to be deployed to resource-constrained edge devices, thus promoting the development of the intelligent world. However, current pruning at initialization methods exhibit poor performance at extreme sparsity. In order to improve the performance of the model under extreme sparsity, this paper proposes a dual-grained lightweight strategy-TEDEPR. This is the first time that TEDEPR has used tensor theory in the pruning at initialization method to optimize the structure of a sparse sub-network model and improve its performance. Specifically, first, at the coarse-grained level, we represent the weight matrix or weight tensor of the model as a low-rank tensor decomposition form and use multi-step chain operations to enhance the feature extraction capability of the base module to construct a low-rank compact network model. Second, unimportant weights are pruned at a fine-grained level based on the trainability of the weights in the low-rank model before the training of the model, resulting in the final compressed model. To evaluate the superiority of TEDEPR, we conducted extensive experiments on MNIST, UCF11, CIFAR-10, CIFAR-100, Tiny-ImageNet and ImageNet datasets with LeNet, LSTM, VGGNet, ResNet and Transformer architectures, and compared with state-of-the-art methods. The experimental results show that TEDEPR has higher accuracy, faster training and inference, and less storage space than other pruning at initialization methods under extreme sparsity.

Treelike process tensor contraction for automated compression of environments

The algorithm “automated compression of environments” (ACE) [M. Cygorek , ] provides a versatile way of simulating an extremely broad class of open quantum systems. This is achieved by encapsulating the influence of the environment, which is determined by the interaction Hamiltonian(s) and initial states, into compact process tensor matrix product operator (PT-MPO) representations. The generality of the ACE method comes at high numerical cost. Here, we demonstrate that orders-of-magnitude improvement of ACE is possible by changing the order of PT-MPO contraction from a sequential to a treelike scheme. The problem of combining two partial PT-MPOs with large inner bonds is solved by a preselection approach. The drawbacks of the preselection approach are that the MPO compression is suboptimal and that it is more prone to error accumulation than sequential combination and compression. We therefore also identify strategies to mitigate these disadvantages by fine-tuning compression parameters. This results in a scheme that is similar in compression efficiency and accuracy to the original ACE algorithm, yet is significantly faster. Our numerical experiments reach similar conclusions for bosonic and fermionic test cases, suggesting that our findings are characteristic of the combination of PT-MPOs more generally. Published by the American Physical Society 2024

Identifying selective PDHK inhibitors using coupled tensor matrix completion and experimental validation

Drug discovery often involves targeting specific members within a family of similar proteins. For example, pyruvate dehydrogenase kinase (PDHK) exists as four isozymes, which exhibit varying expression patterns across multiple tissues. Different PDHK isozymes have been implicated in conditions such as cancer, heart failure, and diabetes, suggesting that targeting them with inhibitors may offer therapeutic benefits. However, simultaneous inhibition of all four PDHK isozymes has the potential to be counterproductive, or poorly tolerated, highlighting a need for isoform-selective PDHK inhibitors. Despite multiple prior reports of PDHK inhibitors, identifying isoform-specific inhibitors for each PDHK isozyme has proven elusive. In this work, we propose a comprehensive framework that combines a machine learning-based prediction method and biochemical testing to screen a library of novel and previously reported compounds, thereby identifying selective inhibitors for PDHK isozymes. Initially, a coupled tensor matrix completion (CTMC) approach is employed to predict compound-target interaction (CTI) and pinpoint target-specific inhibitors. Subsequently, biochemical testing is performed to validate the predicted CTIs. Utilizing this approach, we successfully identified five novel PDHK1-specific inhibitors, underscoring the reliability of this approach as a screening method in the early stages of drug discovery and hit identification.

Investigation of pion-nucleon contributions to nucleon matrix elements

We investigate contributions of excited states to nucleon matrix elements computed in lattice QCD by employing, in addition to the standard nucleon interpolating operator, pion-nucleon (π−N) operators. We solve a generalized eigenvalue problem (GEVP) to obtain an optimal interpolating operator that minimizes overlap with the π−N states. We derive a variant of the standard application of the GEVP method, which allows for constructing 3-point correlation functions using the optimized interpolating operator without requiring the computationally demanding combination that includes π−N operators in both sink and source. We extract nucleon matrix elements using two twisted mass fermion ensembles, one ensemble generated using pion mass of 346 MeV and one ensemble tuned to reproduce the physical value of the pion mass. Especially, we determine the isoscalar and isovector scalar, pseudoscalar, vector, axial, and tensor matrix elements. We include results obtained using a range of kinematic setups, including momentum in the sink. Our results using this variational approach are compared with previous results obtained using the same ensembles and multistate fits without GEVP improvement. We find that for the physical mass point ensemble, the improvement, in terms of suppression of excited states using this method, is most significant for the case of the matrix elements of the isovector axial and pseudoscalar currents. Published by the American Physical Society 2024

Achieving High Piezoelectric Performance Across Multidomains Design in K(Ta,Nb)O3-Based Lead-free Single Crystals.

In light of environmental issues, the lead-free ferroelectric single crystal K(Ta1-xNbx)O3 (KTN) has attracted considerable interest due to its excellent piezoelectric properties. This research focus on improving the piezoelectric performance of KTN by engineering domains through polarization in the [011]c direction. The full-matrix parameters of KTN crystals after polarization along the [011] direction were determined. The finding reveal that KTN polarized in the nonspontaneous polarization direction [011]c exhibits higher piezoelectric constants (d33 = 156 pC/N, d15 = 155 pC/N) and dielectric constants (ε33T ∼ 2337). Additionally, tensor matrix transformation techniques were utilized to analyze the directional dependence of the piezoelectric (d33*), dielectric (ε33T*), elastic (s33E*) constants, and electromechanical coupling coefficient (k33*). The results indicate that these constants reach their maximum values along the polarized [011]c direction. A comparison of the experimental polarization values in the [011]c direction with those calculated for a single-domain state, demonstrates that ferroelectric domains in KTN crystals contribute significantly to the piezoelectric properties. Finally, polarized light microscopy was used to conduct a detailed analysis of the domain structure in KTN single crystals, revealing various orientations and sizes of domain walls. This research offers strategies for optimizing piezoelectric performance.

A novel dimension reduction model based on POD and two-grid Crank–Nicolson mixed finite element methods for 3D nonlinear elastodynamic sine–Gordon problem

Earthquake, petroleum and gas extraction, and underground nuclear test all could cause nonlinear tectonic deformation. It is of great significance to effectively predict and evaluate the disasters and impacts of these geological structure deformation. In this end, a three-dimension (3D) nonlinear elastodynamic sine–Gordon model that can be used to describe nonlinear tectonic deformation including 3D displacement vector, 3 × 3 symmetric stress tensor matrix, nonlinear sin term, and singular initial value functions is first proposed. Then, a new time semi-discrete mixed Crank–Nicolson (TSDMCN) scheme for the 3D nonlinear elastodynamic sine–Gordon model is developed, and the existence, stability, and error estimates of the TSDMCN solutions are proved. Next, a new two-grid mixed finite element Crank–Nicolson (TGMFECN) method with unconditional stability for the 3D nonlinear elastodynamic sine–Gordon model is developed, and the existence, stability, and error estimates of the TGMFECN solutions are proved. Thenceforth, it is the most important thing is that a novel reduced-dimensional iterative TGMFECN (RDITGMFECN) method in matrix form is established by resorting to proper orthogonal decomposition only to lower the unknown TGMFECN solution coefficient vectors and keep TGMFECN basis functions unchanged, which can ensure that the RDITGMFECN method has the same accuracy as the usual TGMFECN method, but can greatly lower the dimension of the unknown TGMFECN solution coefficient vectors so as to mitigate calculated workload, save CPU operating-time, improve computing efficiency, and improve real-time calculating accuracy. In theory, the existence, stability, and error estimates of RDITGMFECN solutions are demonstrated by matrix analysis such that the theoretical analysis becomes very intuitive and easy to be understood by the public, which is a new attempt of theoretical analysis. In application, two numerical examples are used to simulate the 3D nonlinear tectonic deformation caused by earthquake and to verify the correctness of our theoretical results and the effectiveness of the RDITGMFECN method.

Rotational symmetries of 3D point clouds using the covariance matrix and higher-order tensors

We prove that, under generic conditions, the covariance matrix of a 3D point cloud with rotational symmetry has a simple eigenvalue, whose associated eigenvector provides the direction of the axis of rotation, and a double eigenvalue. The direction of the axis of rotation can also be computed from higher order tensors related to the point cloud, which is useful in pathological cases. This leads to a very simple algorithm for detecting rotational symmetry and computing the axis of rotation.

Mechanical mechanism of dip effect on bearing capacity of the pillar strength

Abstract The dip effect on pillar strength in underground ore-body mining is well established, but the variation in stress path (magnitude and direction of stress) due to changing inclination angles requires further study. Using elasticity theory, the Euclidean mean stress tensor characterizes the stress state in pillar zones. Numerical simulations provided the second-order tensor of peak stress for each pillar unit. Through tensor statistical analysis, the Euclidean mean stress tensor matrix was calculated, and its eigenvalues and eigenvectors, representing the magnitude and direction of the principal stress, were derived. This analysis explained the intrinsic dip effect on pillar strength through principal stress characteristics. Finally, the pillar strength envelope function for varying width-height ratios at any dip angle was obtained using the random gradient descent algorithm. Results indicate that in the peak stress state, the average principal stress directions of the pillar change with orebody dip angle, affecting the stress path. The average principal stress increases with pillar size due to increased constraints. These findings offer theoretical insights for pillar design and stability analysis.

Nonrelativistic nuclear reduction for tensor couplings in dark matter direct detection and μ→e conversion

The nonrelativistic effective field theory (NRET) is widely used in dark matter direct detection and charged-lepton flavor violation studies through μ→e conversion. However, existing literature has not fully considered tensor couplings. This study fills this gap by utilizing an innovative tensor decomposition method, extending NRET to incorporate previously overlooked tensor interactions. This development is expected to have a significant impact on ongoing experiments seeking physics beyond the Standard Model and on our understanding of the new-physics interactions. Notably, we identify additional operators in μ→e conversion that are absent in scalar and vector couplings. To support further research and experimental analyses, comprehensive tables featuring tensor matrix elements and their corresponding operators are provided. Published by the American Physical Society 2024

A Comparison of Skeletal Muscle Diffusion Tensor Imaging Tractography Seeding Methods.

The internal arrangement of a muscle's fibers with respect to its mechanical line of action (muscle architecture) is a major determinant of muscle function. Muscle architecture can be quantified using diffusion tensor magnetic resonance imaging-based tractography, which propagates streamlines from a set of seed points by integrating vectors that represent the direction of greatest water diffusion (and by inference, the local fiber orientation). Previous work has demonstrated that tractography outcomes are sensitive to the method for defining seed points, but this sensitivity has not been fully examined. To do so, we developed a realistic simulated muscle architecture and implemented four novel methods for tract seeding: seeding along the muscle-aponeurosis boundary with an updated procedure for rounding seed points prior to lookup in the muscle boundary mask and diffusion tensor matrix (APO-3); voxel-based seeding throughout the muscle volume at a user-specified spatial frequency (VXL-1); voxel-based seeding throughout the muscle volume at a variable spatial frequency (VXL-2), and seeding near external and internal muscle boundaries (VXL-3). We then implemented these methods in an example human dataset. The updated aponeurosis seeding procedures allow more accurate and robust tract propagation from seed points. The voxel-based seeding methods had quantification outcomes that closely matched the updated aponeurosis seeding method. Further, the voxel-based methods can accelerate the overall workflow and may be beneficial in high throughput analysis of multi-muscle datasets. Continued evaluation of these methods in a wider range of muscle architectures is warranted.

ACE: A general-purpose non-Markovian open quantum systems simulation toolkit based on process tensors.

We describe a general-purpose computational toolkit for simulating open quantum systems, which provides numerically exact solutions for composites of zero-dimensional quantum systems that may be strongly coupled to multiple, quite general non-Markovian environments. It is based on process tensor matrix product operators (PT-MPOs), which efficiently encapsulate environment influences. The code features implementations of several PT-MPO algorithms, in particular Automated Compression of Environments for general environments comprised of independent modes as well as schemes for generalized spin boson models. The latter includes a divide-and-conquer scheme for periodic PT-MPOs, which enable million time step simulations for realistic models. PT-MPOs can be precalculated and reused for efficiently probing different time-dependent system Hamiltonians. They can also be stacked together and combined to provide numerically complete solutions of small networks of open quantum systems. The code is written in C++ and is fully controllable by configuration files, for which we have developed a versatile and compact human-readable format.

A reduced-dimension method for unknown Crank-Nicolson finite element solution coefficient vectors of elastic wave equation with singular source term

Herein, we mainly develop a new reduced-dimension method for the unknown finite element (FE) solution coefficient vectors to the elastic wave equation containing the known singular source term, the unknown displacement vector, and the unknown stress tensor matrix. For this end, we need first to develop the Crank-Nicolson (CN) FE (CNFE) method with unconditional stability, unconditional convergence, and second-order time accuracy for the elastic wave equation and analyze the existence, unconditional stability, and errors of the CNFE solutions. After that we employ a proper orthogonal decomposition to lower the dimensionality of the unknown FE solution coefficient vectors in the CNFE method for the elastic wave equation and to design a new reduced-dimension extrapolated CNFE (RDECNFE) method. Subsequently, we employ matrix analysis to discuss the existence, unconditional stability, unconditional convergence, and errors of the RDECNFE solutions. Finally, we use two numerical examples to show the superiority of the RDECNFE method and our theoretical results. It is also shown that the RDECNFE method is feasible and effective for solving the elastic wave equation.

EEG functional connectivity as a Riemannian mediator: An application to malnutrition and cognition.

Mediation analysis assesses whether an exposure directly produces changes in cognitive behavior or is influenced by intermediate "mediators". Electroencephalographic (EEG) spectral measurements have been previously used as effective mediators representing diverse aspects of brain function. However, it has been necessary to collapse EEG measures onto a single scalar using standard mediation methods. In this article, we overcome this limitation and examine EEG frequency-resolved functional connectivity measures as a mediator using the full EEG cross-spectral tensor (CST). Since CST samples do not exist in Euclidean space but in the Riemannian manifold of positive-definite tensors, we transform the problem, allowing for the use of classic multivariate statistics. Toward this end, we map the data from the original manifold space to the Euclidean tangent space, eliminating redundant information to conform to a "compressed CST." The resulting object is a matrix with rows corresponding to frequencies and columns to cross spectra between channels. We have developed a novel matrix mediation approach that leverages a nuclear norm regularization to determine the matrix-valued regression parameters. Furthermore, we introduced a global test for the overall CST mediation and a test to determine specific channels and frequencies driving the mediation. We validated the method through simulations and applied it to our well-studied 50+-year Barbados Nutrition Study dataset by comparing EEGs collected in school-age children (5-11 years) who were malnourished in the first year of life with those of healthy classmate controls. We hypothesized that the CST mediates the effect of malnutrition on cognitive performance. We can now explicitly pinpoint the frequencies (delta, theta, alpha, and beta bands) and regions (frontal, central, and occipital) in which functional connectivity was altered in previously malnourished children, an improvement to prior studies. Understanding the specific networks impacted by a history of postnatal malnutrition could pave the way for developing more targeted and personalized therapeutic interventions. Our methods offer a versatile framework applicable to mediation studies encompassing matrix and Hermitian 3D tensor mediators alongside scalar exposures and outcomes, facilitating comprehensive analyses across diverse research domains.

Towards Super Compressed Neural Networks for Object Identification: Quantized Low-Rank Tensor Decomposition with Self-Attention

Deep convolutional neural networks have a large number of parameters and require a significant number of floating-point operations during computation, which limits their deployment in situations where the storage space is limited and computational resources are insufficient, such as in mobile phones and small robots. Many network compression methods have been proposed to address the aforementioned issues, including pruning, low-rank decomposition, quantization, etc. However, these methods typically fail to achieve a significant compression ratio in terms of the parameter count. Even when high compression rates are achieved, the network’s performance is often significantly deteriorated, making it difficult to perform tasks effectively. In this study, we propose a more compact representation for neural networks, named Quantized Low-Rank Tensor Decomposition (QLTD), to super compress deep convolutional neural networks. Firstly, we employed low-rank Tucker decomposition to compress the pre-trained weights. Subsequently, to further exploit redundancies within the core tensor and factor matrices obtained through Tucker decomposition, we employed vector quantization to partition and cluster the weights. Simultaneously, we introduced a self-attention module for each core tensor and factor matrix to enhance the training responsiveness in critical regions. The object identification results in the CIFAR10 experiment showed that QLTD achieved a compression ratio of 35.43×, with less than 1% loss in accuracy and a compression ratio of 90.61×, with less than a 2% loss in accuracy. QLTD was able to achieve a significant compression ratio in terms of the parameter count and realize a good balance between compressing parameters and maintaining identification accuracy.

White Matter Fiber Tracking Method with Adaptive Correction of Tracking Direction.

The deterministic fiber tracking method has the advantage of high computational efficiency and good repeatability, making it suitable for the noninvasive estimation of brain structural connectivity in clinical fields. To address the issue of the current classical deterministic method tending to deviate in the tracking direction in the region of crossing fiber region, in this paper, we propose an adaptive correction-based deterministic white matter fiber tracking method, named FTACTD. The proposed FTACTD method can accurately track white matter fibers by adaptively adjusting the deflection direction strategy based on the tensor matrix and the input fiber direction of adjacent voxels. The degree of correction direction changes adaptively according to the shape of the diffusion tensor, mimicking the actual tracking deflection angle and direction. Furthermore, both forward and reverse tracking techniques are employed to track the entire fiber. The effectiveness of the proposed method is validated and quantified using both simulated and real brain datasets. Various indicators such as invalid bundles (IB), valid bundles (VB), invalid connections (IC), no connections (NC), and valid connections (VC) are utilized to assess the performance of the proposed method on simulated data and real diffusion-weighted imaging (DWI) data. The experimental results of the simulated data show that the FTACTD method tracks outperform existing methods, achieving the highest number of VB with a total of 13 bundles. Additionally, it identifies the least number of incorrect fiber bundles, with only 32 bundles identified as wrong. Compared to the FACT method, the FTACTD method reduces the number of NC by 36.38%. In terms of VC, the FTACTD method surpasses even the best performing SD_Stream method among deterministic methods by 1.64%. Extensive in vivo experiments demonstrate the superiority of the proposed method in terms of tracking more accurate and complete fiber paths, resulting in improved continuity. The FTACTD method proposed in this study indicates superior tracking results and provides a methodological basis for the investigating, diagnosis, and treatment of brain disorders associated with white matter fiber deficits and abnormalities.

On the Poisson structure and action-angle variables for the Fokas-Lenells equation

In this paper, we employ the inverse scattering transform to investigate the action-angle variables of the Fokas-Lenells equation. Firstly, the Fokas-Lenells equation is derived by the variational principle, and the definition of the Poisson structure is presented. Then, the Poisson brackets between the scattering data are successfully determined by introducing the matrix tensor product. Thus the action-angle variables can be constructed by scattering data. Furthermore, the Hamiltonian of the Fokas-Lenells equation in terms of the scattering data is presented, and the Hamiltonian formalism of the equation is derived.

CS2DIPs: Unsupervised HSI Super-Resolution Using Coupled Spatial and Spectral DIPs.

In recent years, fusing high spatial resolution multispectral images (HR-MSIs) and low spatial resolution hyperspectral images (LR-HSIs) has become a widely used approach for hyperspectral image super-resolution (HSI-SR). Various unsupervised HSI-SR methods based on deep image prior (DIP) have gained wide popularity thanks to no pre-training requirement. However, DIP-based methods often demonstrate mediocre performance in extracting latent information from the data. To resolve this performance deficiency, we propose a coupled spatial and spectral deep image priors (CS2DIPs) method for the fusion of an HR-MSI and an LR-HSI into an HR-HSI. Specifically, we integrate the nonnegative matrix-vector tensor factorization (NMVTF) into the DIP framework to jointly learn the abundance tensor and spectral feature matrix. The two coupled DIPs are designed to capture essential spatial and spectral features in parallel from the observed HR-MSI and LR-HSI, respectively, which are then used to guide the generation of the abundance tensor and spectral signature matrix for the fusion of the HSI-SR by mode-3 tensor product, meanwhile taking some inherent physical constraints into account. Free from any training data, the proposed CS2DIPs can effectively capture rich spatial and spectral information. As a result, it exhibits much superior performance and convergence speed over most existing DIP-based methods. Extensive experiments are provided to demonstrate its state-of-the-art overall performance including comparison with benchmark peer methods.

Decomposable of positive map from M3(C) to M2(M2(C))

In most literature, the decomposition of positive maps from M3 to M2 are discussed where the matrix elements are complex numbers. In this paper we construct a positive maps φ(µ,c1,c2) from M3(C) to M2(M2(C)). The Choi matrices for complete positivity and complete copositivity ares visualized as tensor matrix M3 ⊗M2 with M2(C) as the entry elements. The construction allow us describe decomposability on positive semidefinite matrices.