Dimensional Tensor Research Articles

Criticality and entanglement in nonunitary quantum circuits and tensor networks of noninteracting fermions

Entanglement dynamics in nonunitary quantum systems have been found to exhibit novel nonequilibrium quantum phase transitions and have attracted tremendous attention with connections to various aspects of quantum information theory. This work establishes, for systems of noninteracting fermions, an exact three-way correspondence between (i) nonunitary (as well as unitary) quantum circuits dynamics in $d$ spatial dimensions; (ii) ($d$+1)-dimensional Gaussian tensor networks, and (iii) fermion systems in ($d$+1) spatial dimensions subject to unitary evolution with static Hamiltonians. Through this correspondence, $d$-dimensional random quantum circuits are connected with the classic subject area of Anderson localization in ($d$+1) spatial dimensions and consequently follow the ``tenfold way'' Altland-Zirnbauer classification. Selected examples of quantum circuits exhibiting entanglement phases and criticalities are used to illustrate these general principles.

Bayesian tensor logistic regression with applications to neuroimaging data analysis of Alzheimer's disease.

Alzheimer's disease (AD) can be diagnosed by utilizing traditional logistic regression models to fit magnetic resonance imaging (MRI) data of brain, which is regarded as a vector of covariates. But its parameter estimation is inefficient and computationally extensive due to ultrahigh dimensionality and complicated structure of MRI data. To overcome this deficiency, this paper proposes a tensor logistic regression model (TLRM) for AD's MRI data by regarding MRI tensor as covariates. Under this framework, a tensor candecomp/parafac (CP) decomposition tool is employed to reduce ultrahigh dimensional tensor to a high dimensional level, a novel Bayesian adaptive Lasso method is developed to simultaneously select important components of tensor and estimate model parameters by incorporating the Plya-Gamma method leading a closed-form likelihood and avoiding the usage of the Metropolis-Hastings algorithm, and Gibbs sampler technique in Markov chain Monte Carlo (MCMC). A tensor's product technique is utilized to optimize the calculation program and speed up the calculation of MCMC. Bayes factor together with the path sampling approach is presented to select tensor rank in CP decomposition. Effectiveness of the proposed method is illustrated on simulation studies and an MRI data analysis.

Numerical analysis of a free boundary problem with non-local obstacles

The paper deals with the obstacle-like minimization problem in the cylindrical domain Ω=D×(−l,l)J(u)=∫Ω|∇u|2dx+2∫Dmax{v(x′),0}dx′, where x=(x′,xn), and v(x′)=∫−llu(x′,xn)dxn. The corresponding Euler–Lagrange equation is Δu(x′,xn)=χ{v>0}(x′)+−∂xnu(x′,−l)+∂xnu(x′,l)χ{v=0}(x′). Due to the non-local nature of the obstacle, the comparison principle does not hold for the minimizers u(x), which makes the problem challenging both analytically and numerically. The standard optimization techniques such as Newton or quasi-Newton’s methods require approximations of the Jacobians that are four dimensional tensors and are prohibitively expensive both in storage and computational time due to the nature of the three dimensional problem. In this paper, a new algorithm that can compute the global minimum is introduced. Non-trivial exact solutions have been constructed; and second order accuracy has been confirmed. Another important contribution is the numerical testing of the comparison principle for functions v(x′), as conjectured by M. Chipot and the second author in Chipot and Mikayelyan (2022).

Combined analysis of satellite and ground data for winter wheat yield forecasting

We built machine learning and image analysis tools in order to forecast winter wheat yield based on a rich multi dimensional tensor of agricultural information spanning different scales. This information consists of satellite multi-band images, local soil samples obtained from national databases, local weather as well as field data from 23 farms cultivating winter wheat in southern Sweden. This is inherently a large multi-scale problem due to the large temporal and spatial variation of the input data. We aggregate the data on spatially averaged features over grids which temporally span a seasonal timeline from seeding to harvest. Data cleaning is performed through interpolation for satellite images due to cloud obstructions.Furthermore data is heavily imbalanced since the amount of satellite information far exceeds that of the ground data. Data variance therefore can be an issue which we counter by using a decision tree approach. We find that the Light Gradient Boosting decision tree trained on 262 input features is able to predict winter wheat yield with 82% accuracy.Subsequently we employ game theory in order to better understand the relational importance of specific input features towards forecasting yield. Specifically we find that some of the most important features towards the resulting predictions are the percent clay and magnesium in the soil. Similarly the most important features from the satellite data are: a) the NORM index (Euclidean distance of all bands) computed in the second week of April, b) the NORM index computed in the middle of May as well as c) the second spectral band from the last week of June.

Adaptive gradients and weight projection based on quantized neural networks for efficient image classification

Quantization of weights and activations has been introduced into optimization methodologies of deep neural networks (DNNs) to address issues like memory consumption and massive computational demands that impede edge applications of neural networks. There are multiple approaches to quantization which can probably all be classified into two categories: uniform quantization and non-uniform quantization. The key difference between these methods is whether unequal quantization intervals are used to match the non-uniform distribution of weights. However, conventional techniques train weights layer by layer to suit previously determined quantization points, which poses difficulty for reaching optimal points. We proposed a framework called Deep Projection (DP) to train quantized weights with adaptive gradients. Gradients and weights are projected to high dimensional training space through projection layers, resulting in more complicated update paths and randomness, which benefit the generalization of models. Renewal of weights in the network is conducted by values composited by self-updating high dimensional tensors, which means that the resulting gradients are softer and more adaptive to network training. We applied our method via a uniform quantization approach, and the results showed improvements even when we limited the first and last layers with lower bit-width. The results could be comparable with the non-uniform quantization method with 4-bit precision. Distributions of weights in different quantized networks are analyzed to display the advantages of our method. Besides, all of these steps can be realized in a conventional training process. There are no limitations to quantization bit-width and function. Namely, our framework is readily implemented with a concise architecture.

Quantum annealing algorithms for Boolean tensor networks

Quantum annealers manufactured by D-Wave Systems, Inc., are computational devices capable of finding high-quality heuristic solutions of NP-hard problems. In this contribution, we explore the potential and effectiveness of such quantum annealers for computing Boolean tensor networks. Tensors offer a natural way to model high-dimensional data commonplace in many scientific fields, and representing a binary tensor as a Boolean tensor network is the task of expressing a tensor containing categorical (i.e., {0, 1}) values as a product of low dimensional binary tensors. A Boolean tensor network is computed by Boolean tensor decomposition, and it is usually not exact. The aim of such decomposition is to minimize the given distance measure between the high-dimensional input tensor and the product of lower-dimensional (usually three-dimensional) tensors and matrices representing the tensor network. In this paper, we introduce and analyze three general algorithms for Boolean tensor networks: Tucker, Tensor Train, and Hierarchical Tucker networks. The computation of a Boolean tensor network is reduced to a sequence of Boolean matrix factorizations, which we show can be expressed as a quadratic unconstrained binary optimization problem suitable for solving on a quantum annealer. By using a novel method we introduce called parallel quantum annealing, we demonstrate that Boolean tensor’s with up to millions of elements can be decomposed efficiently using a DWave 2000Q quantum annealer.

Embedding classical dynamics in a quantum computer

We develop a framework for simulating measure-preserving, ergodic dynamical systems on a quantum computer. Our approach provides an operator-theoretic representation of classical dynamics by combining ergodic theory with quantum information science. The resulting quantum embedding of classical dynamics (QECD) enables efficient simulation of spaces of classical observables with exponentially large dimension using a quadratic number of quantum gates. The QECD framework is based on a quantum feature map that we introduce for representing classical states by density operators on a reproducing kernel Hilbert space, $\mathcal{H}$. Furthermore, an embedding of classical observables into self-adjoint operators on $\mathcal{H}$ is established, such that quantum mechanical expectation values are consistent with pointwise function evaluation. In this scheme, quantum states and observables evolve unitarily under the lifted action of Koopman evolution operators of the classical system. Moreover, by virtue of the reproducing property of $\mathcal{H}$, the quantum system is pointwise-consistent with the underlying classical dynamics. To achieve a quantum computational advantage, we project the state of the quantum system onto a finite-rank density operator on a ${2}^{n}$-dimensional tensor product Hilbert space associated with $n$ qubits. By employing discrete Fourier-Walsh transforms of spectral functions, the evolution operator of the finite-dimensional quantum system is factorized into tensor product form, enabling implementation through an $n$-channel quantum circuit of size $O(n)$ and no interchannel communication. Furthermore, the circuit features a state preparation stage, also of size $O(n)$, and a quantum Fourier transform stage of size $O({n}^{2})$, which makes predictions of observables possible by measurement in the standard computational basis. We prove theoretical convergence results for these predictions in the large-qubit limit, $n\ensuremath{\rightarrow}\ensuremath{\infty}$. In light of these properties, QECD provides a consistent simulator of the evolution of classical observables, realized through projective quantum measurement, which is able to simulate spaces of classical observables of dimension ${2}^{n}$ using circuits of size $O({n}^{2})$. We demonstrate the consistency of the scheme in prototypical dynamical systems involving periodic and quasiperiodic oscillators on tori. These examples include simulated quantum circuit experiments in Qiskit Aer, as well as actual experiments on the IBM Quantum System One.

Patch-based Medical Image Segmentation using Matrix Product State Tensor Networks

Tensor networks are efficient factorisations of high dimensional tensors into network of lower order tensors. They have been most commonly used to model entanglement in quantum many-body systems and more recently are witnessing increased applications in supervised machine learning. In this work, we formulate image segmentation in a supervised setting with tensor networks. The key idea is to first lift the pixels in image patches to exponentially high dimensional feature spaces and using a linear decision hyper-plane to classify the input pixels into foreground and background classes. The high dimensional linear model itself is approximated using the matrix product state (MPS) tensor network. The MPS is weight-shared between the non-overlapping image patches resulting in our <em>strided tensor network</em> model. The performance of the proposed model is evaluated on three three 2D- and one 3D- biomedical imaging datasets. The performance of the proposed tensor network segmentation model is compared with relevant baseline methods. In the 2D experiments, the tensor network model yeilds competitive performance compared to the baseline methods while being more resource efficient.

Three-dimensional wind velocity reconstruction based on tensor decomposition and CFD data with experimental verification

Wind energy, which has many advantages, is in a stage of rapid development. However, because wind is sporadic and random, it is difficult to ensure a stable and efficient power supply, which poses risks to the security and stability of the power system. Therefore, research on short-term wind prediction is of great importance. Previous forecasting methods based on vectors or matrices have only been applied to wind velocity distributions in two-dimensional planes. If applied to multiple planes in three-dimensional (3D) space, these methods may not accurately reflect wind velocity distributions. To address this, we propose a novel method of wind forecasting: a tensor-based method that combines Tucker decomposition and computational fluid dynamics (CFD) to rapidly reconstruct 3D wind velocity distributions. A fourth-order wind velocity tensor database under three terrains is established by CFD simulation, then dimensionality reduction and feature extraction are carried out on the database by Tucker decomposition. The coefficient tensors obtained by decomposition are used to rapidly reconstruct 3D wind velocity distributions. Wind fields are successfully reconstructed with good accuracy for direction angles ranging from 0° to 180° and inlet speeds ranging from 0 to 33 m/s. The influences of core tensor dimension, the number and distribution of sensors, and noise on reconstruction error are discussed in the error analysis. Ultimately, the proposed method is verified by anemometer values from a wind tunnel experiment. The minimum relative reconstruction error is 1.79%. The experimental results show that the proposed method can accurately reconstruct 3D wind velocity distributions in wind fields and is an innovative method of short-term wind forecasting.

Tensor denoising with trend filtering

We extend the notion of trend filtering to tensors by considering the $k\mathrm{th}$-order Vitali variation – a discretized version of the integral of the absolute value of the $k\mathrm{th}$-order total derivative. We prove adaptive $\ell^0$-rates and not-so-slow $\ell^1$-rates for tensor denoising with trend filtering. For $k={1,2,3,4}$ we prove that the $d$-dimensional margin of a $d$-dimensional tensor can be estimated at the $\ell^0$-rate $n^{-1}$, up to logarithmic terms, if the underlying tensor is a product of $(k-1)\mathrm{th}$-order polynomials on a constant number of hyperrectangles. For general $k$ we prove the $\ell^1$-rate of estimation $n^{- \frac{H(d)+2k-1}{2H(d)+2k-1}}$, up to logarithmic terms, where $H(d)$ is the $d\mathrm{th}$ harmonic number. Thanks to an ANOVA-type of decomposition we can apply these results to the lower dimensional margins of the tensor to prove bounds for denoising the whole tensor. Our tools are interpolating tensors to bound the effective sparsity for $\ell^0$-rates, mesh grids for $\ell^1$-rates and, in the background, the projection arguments by Dalalyan, Hebiri, and Lederer (2017).

On spectral distribution of sample covariance matrices from large dimensional and large k-fold tensor products

We study the eigenvalue distributions for sums of independent rank-one k-fold tensor products of large n-dimensional vectors. Previous results in the literature assume that k=o(n) and show that the eigenvalue distributions converge to the celebrated Marčenko-Pastur law under appropriate moment conditions on the base vectors. In this paper, motivated by quantum information theory, we study the regime where k grows faster, namely k=O(n). We show that the moment sequences of the eigenvalue distributions have a limit, which is different from the Marčenko-Pastur law, and the Marčenko-Pastur law limit holds if and only if k=o(n) for this tensor model. The approach is based on the method of moments.

Polarimetric spatio-temporal light transport probing

Light emitted from a source into a scene can undergo complex interactions with multiple scene surfaces of different material types before being reflected towards a detector. During this transport, every surface reflection and propagation is encoded in the properties of the photons that ultimately reach the detector, including travel time, direction, intensity, wavelength and polarization. Conventional imaging systems capture intensity by integrating over all other dimensions of the incident light into a single quantity, hiding this rich scene information in these aggregate measurements. Existing methods are capable of untangling these measurements into their spatial and temporal dimensions, fueling geometric scene understanding tasks. However, examining polarimetric material properties jointly with geometric properties is an open challenge that could enable unprecedented capabilities beyond geometric scene understanding, allowing for material-dependent scene understanding and imaging through complex transport, such as macroscopic scattering.In this work, we close this gap, and propose a computational light transport imaging method that captures the spatially- and temporally-resolved complete polarimetric response of a scene, which encodes rich material properties. Our method hinges on a novel 7D tensor theory of light transport. We discover low-rank structure in the polarimetric tensor dimension and propose a data-driven rotating ellipsometry method that learns to exploit redundancy of polarimetric structure. We instantiate our theory with two imaging prototypes: spatio-polarimetric imaging and coaxial temporal-polarimetric imaging. This allows us, for the first time, to decompose scene light transport into temporal, spatial, and complete polarimetric dimensions that unveil scene properties hidden to conventional methods. We validate the applicability of our method on diverse tasks, including shape reconstruction with subsurface scattering, seeing through scattering media, untangling multi-bounce light transport, breaking metamerism with polarization, and spatio-polarimetric decomposition of crystals.

Tensor local linear embedding with global subspace projection optimisation

In this paper, a novel tensor dimensionality reduction (TDR) approach is proposed, which maintains the local geometric structure of tensor data by tensor local linear embedding and explores the global feature by optimising global subspace projection. Firstly, we analyse the local linear feature of tensor data for learning the linear separable embedding of the tenor data. Furthermore, a global subspace projection distance minimisation strategy is introduced to extract the global characteristic of the tensor data. The aim of this strategy is to find an optimal low-dimensional subspace for TDR. In particular, two novel TDR algorithms are developed by the ensemble of tensor local feature preservation and global subspace projection distance minimisation, which express the subspace projection optimisation as an iteration optimisation problem and a Rayleigh quotient problem, respectively. The extensive experimental results on tensor data classification and clustering have demonstrated the proposed algorithms performed well.

Dimensionality reduction of tensor data based on local linear embedding and mode product

There are three contributions in this paper. (1) A tensor version of LLE (short for Local Linear Embedding algorithm) is deduced and presented. LLE is the most famous manifold learning algorithm. Since its proposal, various improvements to LLE have kept emerging without interruption. However, all these achievements are only suitable for vector data, not tensor data. The proposed tensor LLE can also be used a bridge for various improvements to LLE to transfer from vector data to tensor data. (2) A framework of tensor dimensionality reduction based on tensor mode product is proposed, in which the mode matrices can be determined according to specific criteria. (3) A novel dimensionality reduction algorithm for tensor data based on LLE and mode product (LLEMP-TDR) is proposed, in which LLE is used as a criterion to determine the mode matrices. Benefiting from local LLE and global mode product, the proposed LLEMP-TDR can preserve both local and global features of high-dimensional tenser data during dimensionality reduction. The experimental results on data clustering and classification tasks demonstrate that our method performs better than 5 other related algorithms published recently in top academic journals.

MONTI: A Multi-Omics Non-negative Tensor Decomposition Framework for Gene-Level Integrative Analysis.

Multi-omics data is frequently measured to enrich the comprehension of biological mechanisms underlying certain phenotypes. However, due to the complex relations and high dimension of multi-omics data, it is difficult to associate omics features to certain biological traits of interest. For example, the clinically valuable breast cancer subtypes are well-defined at the molecular level, but are poorly classified using gene expression data. Here, we propose a multi-omics analysis method called MONTI (Multi-Omics Non-negative Tensor decomposition for Integrative analysis), which goal is to select multi-omics features that are able to represent trait specific characteristics. Here, we demonstrate the strength of multi-omics integrated analysis in terms of cancer subtyping. The multi-omics data are first integrated in a biologically meaningful manner to form a three dimensional tensor, which is then decomposed using a non-negative tensor decomposition method. From the result, MONTI selects highly informative subtype specific multi-omics features. MONTI was applied to three case studies of 597 breast cancer, 314 colon cancer, and 305 stomach cancer cohorts. For all the case studies, we found that the subtype classification accuracy significantly improved when utilizing all available multi-omics data. MONTI was able to detect subtype specific gene sets that showed to be strongly regulated by certain omics, from which correlation between omics types could be inferred. Furthermore, various clinical attributes of nine cancer types were analyzed using MONTI, which showed that some clinical attributes could be well explained using multi-omics data. We demonstrated that integrating multi-omics data in a gene centric manner improves detecting cancer subtype specific features and other clinical features, which may be used to further understand the molecular characteristics of interest. The software and data used in this study are available at: https://github.com/inukj/MONTI.

Copositivity for 3rd-Order Symmetric Tensors and Applications

The strict opositivity of 4th order symmetric tensor may apply to detect vacuum stability of general scalar potential. For finding analytical expressions of (strict) opositivity of 4th order symmetric tensor, we may reduce its order to 3rd order to better deal with it. So, it is provided that several analytically sufficient conditions for the copositivity of 3th order 2 dimensional (3 dimensional) symmetric tensors. Subsequently, applying these conclusions to 4th order tensors, the analytically sufficient conditions of copositivity are proved for 4th order 2 dimensional and 3 dimensional symmetric tensors. Finally, we apply these results to present analytical vacuum stability conditions for vacuum stability for $\mathbb{Z}_3$ scalar dark matter.

Quantum tensor singular value decomposition* *This research is supported in part by Hong Kong Research Grant council (RGC) grants (No. 15208418, No. 15203619, No. 15506619) and Shenzhen Fundamental Research Fund, China under Grant No. JCYJ20190813165207290.

Tensors are increasingly ubiquitous in various areas of applied mathematics and computing, and tensor decompositions are of practical significance and benefit many applications in data completion, image processing, computer vision, collaborative filtering, etc. Recently, Kilmer and Martin propose a new tensor factorization strategy, tensor singular value decomposition (t-svd), which extends the matrix singular value decomposition to tensors. However, computing t-svd for high dimensional tensors costs much computation and thus cannot efficiently handle large scale datasets. Motivated by advantage of quantum computation, in this paper, we present a quantum algorithm of t-svd for third-order tensors and then extend it to order-p tensors. We prove that our quantum t-svd algorithm for a third-order N dimensional tensor runs in time if we do not recover classical information from the quantum output state. Moreover, we apply our quantum t-svd algorithm to context-aware multidimensional recommendation systems, where we just need to extract partial classical information from the quantum output state, thus achieving low time complexity.

Tensor dimensionality reduction via mode product and HSIC

Tensor dimensionality reduction (TDR) is a hot research topic in machine learning, which learns data representations by preserving the original data structure while avoiding convert samples into vectors and solving the problem of the curse of dimensionality of tensor data. In the work, a novel TDR approach based on mode product and Hilbert–Schmidt Independence criterion (HSIC) is proposed. The contributions of authors' work is described as following: (1) HSIC measures the statistical correlation of two random variables. However, instead of measuring the statistical correlation of two random variables directly, HSIC first transforms the two random variables into two reproducing kernel Hilbert spaces (RKHSs), and then measures the statistical correlation of transformed random variables by using Hilbert–Schmidt operators between the two RKHSs. The exploitation of RKHS increases the flexibility and applicability of HSIC. Although HSIC is widely used in machine learning, the authors have not seen its application to dimensionality reduction (DR)(except for authors' previous work). (2) A novel HSIC-based TDR approach is proposed, which first applies HSIC to capture statistical information of tensor data set for DR. The authors give the mathematical derivation of HSIC for tensor data and establish a framework of TDR based on HSIC, named HSIC-TDR for short, which aims to improve the DR results of tensor by exploring and preserving the statistical information of original data set. (3) Furthermore, to solve the out-of-sample problem, the authors learn an explicit expression between the dimensionality-reduced tensors and the higher-dimensional tensors by introducing mode product to HSIC-TDR. The experimental results between the proposed method and other state-of-the-art algorithm on various datasets demonstrate the well performance of the proposed method.

Hybrid CUR-type decomposition of tensors in the Tucker format

The paper introduces a hybrid approach to the CUR-type decomposition of tensors in the Tucker format. The idea of the hybrid algorithm is to write a tensor $\mathcal{X}$ as a product of a core tensor $\mathcal{S}$, a matrix $C$ obtained by extracting mode-$k$ fibers of $\mathcal{X}$, and matrices $U_j$, $j=1,\ldots,k-1,k+1,\ldots,d$, chosen to minimize the approximation error. The approximation can easily be modified to preserve the fibers in more than one mode. The approximation error obtained this way is smaller than the one from the standard tensor CUR-type method. This difference increases as the tensor dimension increases. It also increases as the number of modes in which the original fibers are preserved decreases.

GFCCLib: Scalable and efficient coupled-cluster Green's function library for accurately tackling many-body electronic structure problems

Coupled-cluster Green's function (GFCC) calculation has drawn much attention in the recent years for targeting the molecular and material electronic structure problems from a many-body perspective in a systematically improvable way. However, GFCC calculations on scientific computing clusters usually suffer from expensive higher dimensional tensor contractions in the complex space, expensive inter-process communication, and severe load imbalance, which limits it's use for tackling electronic structure problems. Here we present a numerical library prototype that is specifically designed for large-scale GFCC calculations. The design of the library is focused on a systematically optimal computing strategy to improve its scalability and efficiency. The performance of the library is demonstrated by the relevant profiling analysis of running GFCC calculations on remote giant computing clusters. The capability of the library is highlighted by computing a wide near valence band of a fullerene C60 molecule for the first time at the GFCCSD level that shows excellent agreement with the experimental spectrum. Program summaryProgram Title: GFCCLibCPC Library link to program files:https://doi.org/10.17632/j594wydctd.1Code Ocean capsule:https://doi.org/10.24433/CO.5131827.v1Licensing provisions: MIT LicenseProgramming language: C++Nature of problem: The applications of coupled cluster Green's function on large scale molecular electronic structure problems suffer from expensive higher dimensional tensor contractions in the complex space, expensive inter-process communication, and severe load imbalance. Tackling these issues are a key step in building high-performance coupled cluster Green's function library for its routine use in large scale molecular science.Solution method: We have developed a C++ library for large scale molecular GFCC calculations on high-performance computing clusters. We provide implementations for high dimensional tensor algebra for many-body methods (TAMM), Cholesky decomposition of high dimensional electron repulsion integral tensors, process group technique for mitigating load imbalance. The library is written in C++. The source code, tutorials and documentation are provided online. A continuous integration mechanism is set up to automatically run a series of regression tests and check code coverage when the codebase is updated.