Tensor Representation Research Articles

Utilizing machine learning and molecular dynamics for enhanced drug delivery in nanoparticle systems

Materials data science and machine learning (ML) are pivotal in advancing cancer treatment strategies beyond traditional methods like chemotherapy. Nanotherapeutics, which merge nanotechnology with targeted drug delivery, exemplify this advancement by offering improved precision and reduced side effects in cancer therapy. The development of these nanotherapeutic agents depends critically on understanding nanoparticle (NP) properties and their biological interactions, often analyzed through molecular dynamics (MD) simulations. This study enhances these analyses by integrating ML with MD simulations, significantly improving both prediction accuracy and computational efficiency. We introduce a comprehensive three-stage methodology for predicting the solvent-accessible surface area (SASA) of NPs, which is crucial for their therapeutic efficacy. The process involves training an ML model to forecast the many-body tensor representation (MBTR) for future time steps, applying data augmentation to increase dataset realism, and refining the SASA predictor with both augmented and original data. Results demonstrate that our methodology can predict SASA values 299 time steps ahead with a 40-fold speed improvement and a 25% accuracy increase over existing methods. Importantly, it provides a 300-fold increase in computational speed compared to traditional simulation techniques, offering substantial cost and time savings for nanotherapeutic research and development.

Trace Ratio Based Manifold Learning With Tensor Data

ABSTRACTIn this article, we propose an extension of trace ratio based Manifold learning methods to deal with multidimensional data sets. Based on recent progress on the tensor‐tensor product, we present a generalization of the trace ratio criterion by using the properties of the t‐product. This will conduct us to introduce some new concepts such as Laplacian tensor and we will study formally the trace ratio problem by discussing the conditions for the existence of solutions and optimality. Next, we will present a tensor Newton QR decomposition algorithm for solving the trace ratio problem. Manifold learning methods such as Laplacian eigenmaps, linear discriminant analysis and locally linear embedding will be formulated in a tensor representation and optimized by the proposed algorithm. Finally, we will evaluate the performance of the different studied dimension reduction methods on several synthetic and real world data sets.

STCO: Enhancing Training Efficiency via Structured Sparse Tensor Compilation Optimization

Network sparsification serves as an effective technique to accelerate Deep Neural Network (DNN) inference. However, existing sparsification techniques often rely on structured sparsity, which yields limited benefits. This is primarily due to the significant memory and computational overhead introduced by numerous sparse storage formats during address generation and gradient updates. Additionally, many of these solutions are tailored solely for the inference phase, neglecting the crucial training phase. In this paper, we introduce STCO, a novel Sparse Tensor Compilation Optimization technique that significantly enhances training efficiency through structured sparse tensor compilation. Central to STCO is the Tensorization-aware Index Entity (TIE) format, which effectively represents structured sparse tensors by eliminating redundant indices and minimizing storage overhead. The TIE format plays a pivotal role in the Address-carry flow (AC flow) pass, which optimizes the data layout at the computational graph level. This pass leverages the TIE format to enhance the efficiency of tensor representations, enabling more compact and efficient sparse tensor storage. Meanwhile, a shape inference pass utilizes the AC flow to derive optimized tensor shapes, further refining the performance of sparse tensor operations. Moreover, the Address-Carry TIE Flow dynamically tracks nonzero addresses, extending the benefits of sparse optimization to both forward and backward propagation. This seamless integration into the training pipeline enables a smooth transition to sparse tensor compilation without significant modifications to existing codebases. To further boost training performance, we implement an operator-level AC flow optimization pass tailored for structured sparse tensors. This pass generates efficient addresses, ensuring minimal computational overhead during sparse tensor operations. The flexibility of STCO allows it to be efficiently integrated into various frameworks or compilers, providing a robust solution for enhancing training efficiency with structured sparse tensors. Experiments demonstrated that STCO achieved impressive speedups of 3.64 ×, 5.43 ×, 4.89 ×, and 3.91 × when compared to state-of-the-art sparse formats on VGG16, ResNet-18, MobileNetV1, and MobileNetV2, respectively. These findings underscore the efficiency and superiority of our proposed approach in leveraging unstructured sparsity for Deep Neural Network inference acceleration.

Time-reversal anomalies of QCD3 and QED3

Anomalies of global symmetry provide powerful tool to constrain the dynamics of quantum systems, such as anomaly matching in the renormalization group flow and obstruction to symmetric mass generation. In this note we compute the anomalies in 2+1d time-reversal symmetric gauge theories with massless fermions in the fundamental and rank-two tensor representations, where the gauge groups are SU(N), SO(N), Sp(N), U(1). The fermion parity is part of the gauge group and the theories are bosonic. The time-reversal symmetry satisfies T2 = 1 or T2 = M\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{M} $$\\end{document} where M\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{M} $$\\end{document} is an internal magnetic symmetry. We show that some of the bosonic gauge theories have time-reversal anomaly with c− ≠ 0 mod 8 that is absent in fermionic systems. The anomalies of the gauge theories can be nontrivial even when the number of Majorana fermions is a multiple of 16 and ν = 0.

Unravelling the Enigmatic Nexus: Black Holes, Dark Matter, and the Interplay of Light, Gravity, and Electromagnetic Forces in Astrophysics and Astronomy

This research introduces a mathematical model positioning our physical reality within a ten-dimensional hyperspace, in which time acts as the unifying coordinate linking three-dimensional electric, magnetic, and gravitational spaces. Each domain is characterized by its respective field—electric, magnetic, or gravitational—and governed by intrinsic divergences and rotations, leading to a universal property known as Force Density, expressed in [N/m³]. This Force Density facilitates interactions among the distinct spaces while adhering to the principle of equilibrium, which posits that cumulative force densities from disturbances must consistently sum to zero. Building upon Einstein's General Relativity—which describes the curvature of spacetime by gravitational fields and assumes a constant light speed—this study proposes a perspective wherein light speed may vary during coherent laser beam interactions, prompting a re-examination of gravitational and luminous interactions across scales. The proposed model integrates the Stress-Energy Tensor and Gravitational Tensor, introducing a new tensor representation for black holes, termed Gravitational Electromagnetic Confinements, incorporating electromagnetic energy gradients and Lorentz transformations. This framework transcends traditional General Relativity, particularly evident in gravitational lensing. By reinterpreting Einstein's incorporation of the Gravitational Constant within the Energy-Stress Tensor, this work harmonizes gravity and light, offering insights into black hole solutions resonating with John Archibald Wheeler's 1955 research. Empirical data from Galileo satellites and MASER frequency measurements underscore discrepancies between established theories and this new model, enhancing the precision of gravitational observations. Through the confluence of Quantum Physics and General Relativity, as seen in approaches like String Theory, this interdisciplinary endeavor revisits the gravitational constant "G," redefining it while bridging theoretical frameworks, thus paving the way for breakthroughs in astronomical and astrophysical sciences.

A new nonconvex multi-view subspace clustering via learning a clean low-rank representation tensor

Abstract Recently, low-rank tensor representation has achieved impressive results for multi-view subspace clustering (MSC). The typical MSC methods utilize the tensor nuclear norm as a convex surrogate of the tensor multi-rank to obtain a low-rank representation, which exhibits limited robustness when dealing with noisy and complex data scenarios. In this paper, we introduce an innovative clean low-rank tensor representation approach that combines the idea of tensor robust principal component analysis (TRPCA) with a new nonconvex tensor multi-rank approximation regularization. This integration enhances the robustness of the low-rank representation, resulting in improved performance. Furthermore, to better capture the local geometric features, we employ a higher-order manifold regularization term. To effectively address our new model, we develop an iterative algorithm that has been proven to converge to the desired Karush-Kuhn-Tucker (KKT) point. The numerical experiments on widely used datasets serve to demonstrate the efficacy and effectiveness of our new method.

E-FNet: A EEG-fNIRS dual-stream model for Brain-Computer Interfaces

With the rapid advancements in Brain-Computer Interfaces (BCI) and multimodal technology, researchers are actively exploring the intricate domain of multimodal BCI. The integration of electroencephalography (EEG) and functional near-infrared spectroscopy (fNIRS) in BCI systems has garnered significant attention due to the combination of strengths from both modalities and the overcoming of limitations inherent in standalone systems. In this study, the E-FNet architecture is introduced, a (2+1)D dual-stream model explicitly designed to process multimodal EEG and fNIRS data for brain-machine interface applications. By leveraging the (2+1)D architecture and adopting a bias-free design, E-FNet efficiently processes multimodal data. A unified 4D tensor representation enables the achievement of both temporal and spatial alignment of EEG and fNIRS signals. For Motor Imagery (MI) and Mental Arithmetic (MA) tasks, the early integration of EEG and fNIRS signals yields remarkable accuracy rates of 95.86% and 95.80% respectively, outperforming the accuracy obtained with EEG signals alone. Moreover, the minimal variability in results across subjects underscores the complementary effects achieved through the integration of different signals.

OQuPy: A Python package to efficiently simulate non-Markovian open quantum systems with process tensors.

Non-Markovian dynamics arising from the strong coupling of a system to a structured environment is essential in many applications of quantum mechanics and emerging technologies. Deriving an accurate description of general quantum dynamics including memory effects is, however, a demanding task, prohibitive to standard analytical or direct numerical approaches. We present a major release of our open source software package, OQuPy (Open Quantum System in Python), which provides several recently developed numerical methods that address this challenging task. It utilizes the process tensor approach to open quantum systems (OQS) in which a single map, the process tensor, captures all possible effects of an environment on the system. The representation of the process tensor in a tensor network form allows for an exact yet highly efficient description of non-Markovian OQS (NM-OQS). The OQuPy package provides methods to (1) compute the dynamics and multi-time correlations of quantum systems coupled to single and multiple environments, (2) optimize control protocols for NM-OQS, (3) simulate interacting chains of NM-OQS, and (4) compute the mean-field dynamics of an ensemble of NM-OQS coupled to a common central system. Our aim is to provide an easily accessible and extensible tool for researchers of OQS in fields such as quantum chemistry, quantum sensing, and quantum information.

Generalized cluster states from Hopf algebras: non-invertible symmetry and Hopf tensor network representation

Cluster states are crucial resources for measurement-based quantum computation (MBQC). It exhibits symmetry-protected topological (SPT) order, thus also playing a crucial role in studying topological phases. We present the construction of cluster states based on Hopf algebras. By generalizing the finite group valued qudit to a Hopf algebra valued qudit and introducing the generalized Pauli-X operator based on the regular action of the Hopf algebra, as well as the generalized Pauli-Z operator based on the irreducible representation action on the Hopf algebra, we develop a comprehensive theory of Hopf qudits. We demonstrate that non-invertible symmetry naturally emerges for Hopf qudits. Subsequently, for a bipartite graph termed the cluster graph, we assign the identity state and trivial representation state to even and odd vertices, respectively. Introducing the edge entangler as controlled regular action, we provide a general construction of Hopf cluster states. To ensure the commutativity of the edge entangler, we propose a method to construct a cluster lattice for any triangulable manifold. We use the 1d cluster state as an example to illustrate our construction. As this serves as a promising candidate for SPT phases, we construct the gapped Hamiltonian for this scenario and provide a detailed discussion of its non-invertible symmetries. We demonstrate that the 1d cluster state model is equivalent to the quasi-1d Hopf quantum double model with one rough boundary and one smooth boundary. We also discuss the generalization of the Hopf cluster state model to the Hopf ladder model through symmetry topological field theory. Furthermore, we introduce the Hopf tensor network representation of Hopf cluster states by integrating the tensor representation of structure constants with the string diagrams of the Hopf algebra, which can be used to solve the Hopf cluster state model.

Dynamical reconstruction of the Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda $$\\end{document}CDM model in hybrid metric-Palatini gravity

In this work, we apply the formalism of dynamical systems to analyze the viability of the Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda $$\\end{document}CDM model in a generalized form of the hybrid metric-Palatini gravity theory written in terms of its dynamically equivalent scalar–tensor representation. Adopting a matter distribution composed of two relativistic fluids described by the equations of state of radiation and pressureless dust, one verifies that the cosmological phase space features the usual curvature-dominated, radiation-dominated, matter-dominated, and exponentially accelerated fixed points, even in the absence of a dark energy component. A numerical integration of the dynamical equations describing the system, subjected to initial conditions consistent with the cosmographic observations from the Planck satellite and weak-field solar system dynamics, shows that cosmological solutions mimicking the behavior of the Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda $$\\end{document}CDM model in General Relativity (GR) are attainable in this theory, with the deviations from GR being exponentially suppressed at early-times and the scalar-field potential effectively playing the role of dark energy at late times.

Equivariant tensor network potentials

Abstract Machine-learning interatomic potentials (MLIPs) have made a significant contribution to the recent progress in the fields of computational materials and chemistry due to the MLIPs’ ability of accurately approximating energy landscapes of quantum-mechanical models while being orders of magnitude more computationally efficient. However, the computational cost and number of parameters of many state-of-the-art MLIPs increases exponentially with the number of atomic features. Tensor (non-neural) networks, based on low-rank representations of high-dimensional tensors, have been a way to reduce the number of parameters in approximating multidimensional functions, however, it is often not easy to encode the model symmetries into them. In this work we develop a formalism for rank-efficient equivariant tensor networks (ETNs), i.e. tensor networks that remain invariant under actions of SO(3) upon contraction. All the key algorithms of tensor networks like orthogonalization of cores and DMRG-based algorithms carry over to our equivariant case. Moreover, we show that many elements of modern neural network architectures like message passing, pulling, or attention mechanisms, can in some form be implemented into the ETNs. Based on ETNs, we develop a new class of polynomial-based MLIPs that demonstrate superior performance over existing MLIPs for multicomponent systems.

A Direct Approach to the Polar Representation of Plane Tensors

We show we can derive the so called polar representation of 2D symmetric second and fourth order real tensors essentially just relying on the spectral theorem for unitary tensors in the complex field. The use of a coordinate-free approach allows us to clearly and methodically detect rotation-invariant quantities and to readily (and directly) deduce the representation theorems.

Joint local smoothness and low-rank tensor representation for robust multi-view clustering

Low-rank tensor representation (LRTR) has become a significant method for achieving improved multi-view clustering (MVC) performance. Generally, most LRTR methods impose a tensor low-rank constraint (TLRC) on a tensor, which is spliced by the representation matrix of each view, to explore the low-rank prior hidden in these representation matrices. However, two problems remain unsolved. On the one hand, these representation matrices still contain noise since the raw data usually possess noise; on the other hand, and more importantly, is there any other prior information that can be explored among these representation matrices? Since samples located in the same cluster are similar, the coefficients of their representation matrices should exhibit similar. That is, these representation matrices should be local smoothness (LS), which is also verified by our numerical tests. In addition, the use of the LS prior helps to denoise the noise contain in the data. Then, in this paper, we propose a joint LS and LRTR for robust MVC method (LS-LRTR), which can simultaneously exploit the low-rank and LS priors. Specifically, we utilize a TLRC to explore the low-rank prior in the representation matrices. Subsequently, to mine the LS prior and further reduce the influence of noise, we introduce the Total Variation (TV) norm to the constraint representation matrices. Then, we fuse the TLRC and TV norm into a unified framework. Additionally, we apply an Augmented Lagrange Multiplier to solve the optimization problem of LS-LRTR. Experiments conducted on several datasets indicate that LS-LRTR outperforms the state-of-the-art clustering methods.

TensoSDF: Roughness-aware Tensorial Representation for Robust Geometry and Material Reconstruction

Reconstructing objects with realistic materials from multi-view images is problematic, since it is highly ill-posed. Although the neural reconstruction approaches have exhibited impressive reconstruction ability, they are designed for objects with specific materials (e.g., diffuse or specular materials). To this end, we propose a novel framework for robust geometry and material reconstruction, where the geometry is expressed with the implicit signed distance field (SDF) encoded by a tensorial representation, namely TensoSDF. At the core of our method is the roughness-aware incorporation of the radiance and reflectance fields, which enables a robust reconstruction of objects with arbitrary reflective materials. Furthermore, the tensorial representation enhances geometry details in the reconstructed surface and reduces the training time. Finally, we estimate the materials using an explicit mesh for efficient intersection computation and an implicit SDF for accurate representation. Consequently, our method can achieve more robust geometry reconstruction, outperform the previous works in terms of relighting quality, and reduce 50% training times and 70% inference time. Codes and datasets are available at https://github.com/Riga2/TensoSDF.

Multi-source partial multi-label learning via tensor decomposition and nonconvex regularization

Multi-source partial multi-label learning (MSPMLL) aims to learn a multi-label classifier from training examples with multi-source features and multiple labels. Most of the existing MSPMLL models tackle each source independently to derive classifiers. Those methods cannot efficiently reveal the intrinsic structure and relationships among instances across different sources, nor deal with inconsistent labels caused by noise. To address those challenges, this paper proposes a tensor decomposition and nonconvex regularization approach to multi-source partial multi-label learning, nominated as MSPML-TLR. In this model, a hypergraph based affinity graph is constructed from each of multi-source data in the feature space. A tensor representation of multi-source multi-label data is proposed based on the affinity graphs. A multi-linear projection is learned to map the feature affinity space to a latent ground-truth label space via tensor robust principal component analysis. Furthermore, a notion called the parameterized tensorial logarithmic rank is designed to approximate the rank of the multi-linear projection and the latent ground-truth label tensor. Moreover, a hypergraph-based Laplacian regularization is imposed on each frontal slice of the latent ground-truth label tensor to preserve the topological structure of instances across multi-source feature spaces. An efficient algorithm is designed to implement MSPML-TLR and the convergence of the algorithm is rigorously proven. Extensive experiments on various real-world benchmark datasets demonstrate that the proposed model outperforms state-of-the-art methods in multi-label classification tasks.

On second-order tensor representation of derivatives in shape optimization.

In this article, we study general properties of distributed shape derivatives admitting a volumetric tensor representation of order two. We obtain a general result providing a range of expressions for the shape derivative, with the distributed shape derivative at one end of the range and the standard Hadamard formula at the other end. We further apply this result to a cost functional depending on the solution of a fourth-order elliptic equation, and obtain the distributed shape derivative in the case of open sets, and the Hadamard formula for sets of class [Formula: see text]. We also consider the case of polygons, for which a description of the weak singularities of the solution appearing in the neighbourhood of the vertices is required to obtain the Hadamard formula. This article is part of the theme issue 'Non-smooth variational problems with applications in mechanics'.

Tensorized Incomplete Multi-view Kernel Subspace Clustering

Recently considerable advances have been achieved in the incomplete multi-view clustering (IMC) research. However, the current IMC works are often faced with three challenging issues. First, they mostly lack the ability to recover the nonlinear subspace structures in the multiple kernel spaces. Second, they usually neglect the high-order relationship in multiple representations. Third, they often have two or even more hyper-parameters and may not be practical for some real-world applications. To tackle these issues, we present a Tensorized Incomplete Multi-view Kernel Subspace Clustering (TIMKSC) approach. Specifically, by incorporating the kernel learning technique into an incomplete subspace clustering framework, our approach can robustly explore the latent subspace structure hidden in multiple views. Furthermore, we impute the incomplete kernel matrices and learn the low-rank tensor representations in a mutual enhancement manner. Notably, our approach can discover the underlying relationship among the observed and missing samples while capturing the high-order correlation to assist subspace clustering. To solve the proposed optimization model, we design a three-step algorithm to efficiently minimize the unified objective function, which only involves one hyper-parameter that requires tuning. Experiments on various benchmark datasets demonstrate the superiority of our approach. The source code and datasets are available at: https://www.researchgate.net/publication/381828300_TIMKSC_20240629.

CoNoT: Coupled Nonlinear Transform-Based Low-Rank Tensor Representation for Multidimensional Image Completion.

Recently, the transform-based tensor nuclear norm (TNN) methods have shown promising performance and drawn increasing attention in tensor completion (TC) problems. The main idea of these methods is to exploit the low-rank structure of frontal slices of the tensor under the transform. However, the transforms in TNN methods usually treat all modes equally and do not consider the different traits of different modes (i.e., spatial and spectral/temporal modes). To address this problem, we suggest a new low-rank tensor representation based on the coupled nonlinear transform (called CoNoT) for a better low-rank approximation. Concretely, spatial and spectral/temporal transforms in the CoNoT, respectively, exploit the different traits of different modes and are coupled together to boost the implicit low-rank structure. Here, we use the convolutional neural network (CNN) as the CoNoT, which can be learned solely from an observed multidimensional image in an unsupervised manner. Based on this low-rank tensor representation, we build a new multidimensional image completion model. Moreover, we also propose an enhanced version (called Ms-CoNoT) to further exploit the spatial multiscale nature of real-world data. Extensive experiments on real-world data substantiate the superiority of the proposed models against many state-of-the-art methods both qualitatively and quantitatively.

Pattern-Based Recommender System Using Nuclear Norm Minimization of Three-Mode Tensor and Quantum Fidelity-Based K-Means

Recommender systems (RSs) consist of predicting missing ratings based on the observed ones. This problem corresponds to matrix completion where users are its rows and items are its columns and it contains the observed ratings. An efficient RS is the one promoting the personal relevancy of its users which is not the case in the matrix completion process. It takes into account all the rates for prediction without including the users’ and items’ characteristics. Patterns are the key enablers of such solutions. In this work, we present a three-mode tensor representation with two aspects namely user–item interactions (observed ratings) and item–item relationship (detected similarities). To capture the similarities, the patterns are grouped in an equivalent manner using a bi-quantum clustering process (Quantum K-means). This step is adopted to consider only the relevant observed ratings in the prediction process and express item-to-item relationship using fidelity distance. Then, for each missing rating r that a user u might give to an item i in the future, a sub-tensor is created according to the detected patterns. This sub-tensor then is completed by minimizing its rank. This problem is NP-hard, hence a surrogate is used which is the nuclear norm. The effectiveness of the proposed approach is measured according to information retrieval evaluation criteria: precision, recall and [Formula: see text]-measure. The proposed approach improved the precision of the state-of-the-art methods by 20[Formula: see text].

Solving Fermi-Hubbard-type models by tensor representations of backflow corrections

Solving Fermi-Hubbard-type models by tensor representations of backflow corrections