Matrix Factorizations Research Articles

Discrete-time analytic signals for power system phasor and frequency tracking

The tracking of phasor and frequency is becoming vital in power networks ranging from interconnections monitored via wide-area measurement systems to isolated microgrids operating autonomously. However, despite the progress in phasor measurement technology, the performance of the signal processing module is still an active research area due to the challenge of phasor and frequency estimation under dynamic conditions. This paper proposes a signal processing technique for frequency and phasor estimation based on discrete-time analytic (DTA) signals. First, a new method is proposed for computing the DTA signal directly in the time domain. The DTA signal is then approximated using a first-order Taylor series expansion that forms the basis of the tracking method, essentially ensuring that the DTA signal is generated at the exact value of the estimated frequency. The proposed method is accurate as it is based on time-domain least squares estimation; it also meets real-time computing requirements because it employs a single factorization of the gain matrix to achieve the solution. Numerical results are reported under both steady-state and dynamic conditions, showing the proposal’s viability.

Learning matrix factorization with scalable distance metric and regularizer

Matrix factorization has always been an encouraging field, which attempts to extract discriminative features from high-dimensional data. However, it suffers from negative generalization ability and high computational complexity when handling large-scale data. In this paper, we propose a learnable deep matrix factorization via the projected gradient descent method, which learns multi-layer low-rank factors from scalable metric distances and flexible regularizers. Accordingly, solving a constrained matrix factorization problem is equivalently transformed into training a neural network with an appropriate activation function induced from the projection onto a feasible set. Distinct from other neural networks, the proposed method activates the connected weights not just the hidden layers. As a result, it is proved that the proposed method can learn several existing well-known matrix factorizations, including singular value decomposition, convex, nonnegative and semi-nonnegative matrix factorizations. Finally, comprehensive experiments demonstrate the superiority of the proposed method against other state-of-the-arts.

Parallel block preconditioners for virtual element discretizations of the time-dependent Maxwell equations

The focus of this study is the construction and numerical validation of parallel block preconditioners for low order virtual element discretizations of the three-dimensional Maxwell equations. The virtual element method (VEM) is a recent technology for the numerical approximation of partial differential equations (PDEs), that generalizes finite elements to polytopal computational grids. So far, VEM has been extended to several problems described by PDEs, and recently also to the time-dependent Maxwell equations. When the time discretization of PDEs is performed implicitly, at each time-step a large-scale and ill-conditioned linear system must be solved, that, in case of Maxwell equations, is particularly challenging, because of the presence of both H(div) and H(curl) discretization spaces. We propose here a parallel preconditioner, that exploits the Schur complement block factorization of the linear system matrix and consists of a Jacobi preconditioner for the H(div) block and an auxiliary space preconditioner for the H(curl) block. Several parallel numerical tests have been performed to study the robustness of the solver with respect to mesh refinement, shape of polyhedral elements, time step size and the VEM stabilization parameter.

Objective Bayesian Approach for Binary Hypothesis Testing of Multivariate Gaussian Observations

This paper proposes an objective Bayesian detector for the binary hypothesis testing problem in which the observation vectors are IID and follow multivariate Gaussian distribution under both the null and alternative hypotheses. The mean vector and covariance matrix under each hypothesis are not known, they also do not have special structure other than the covariance matrices are partially ordered in the positive definite cone with their difference positive semi-definite. An example of such a problem is the detection of Gaussian random signal in Gaussian noise. Non-informative priors are imposed on the unknown parameters for computing the marginals and evaluating the Bayes factor test statistic, where the prior for the covariance matrix is applied to the Cholesky factorization of the precision matrix. For the null hypothesis, we propose conjugate priors for the unknowns. For the alternative hypothesis, we propose uniform prior on the mean vector, and a class of objective priors that encompasses the Jeffreys, Independence Jeffreys, Geisser and Cornfield, Haar Measure and Reference priors for the precision matrix. The proposed priors enable closed-form expressions for the marginals and lead to an objective Bayes factor having the posteriors mainly governed by the data. The proposed detector is analyzed for justifying the priors used, deriving the theoretical moments, and approximating the false alarm and detection probability densities by the shifted Gamma distributions. The developed detector exhibits improvement in detection performance over the energy detector (ED) and the generalized likelihood ratio test (GLRT).

A flexible block classical Gram–Schmidt skeleton with reorthogonalization

AbstractWe investigate a variant of the reorthogonalized block classical Gram–Schmidt method for computing the QR factorization of a full column rank matrix. Our aim is to bound the loss of orthogonality even when the first local QR algorithm is only conditionally stable. In particular, this allows the use of modified Gram–Schmidt instead of Householder transformations as the first local QR algorithm. Numerical experiments confirm the stable behavior of the new variant. We also examine the use of non‐QR local factorization and show by example that the resulting variants, although less stable, may also be applied to ill‐conditioned problems.

On Computing with Some Convex Relaxations for the Maximum-Entropy Sampling Problem

Based on a factorization of an input covariance matrix, we define a mild generalization of an upper bound of Nikolov and of Li and Xie for the NP-hard constrained maximum-entropy sampling problem ( CMESP ). We demonstrate that this factorization bound is invariant under scaling and independent of the particular factorization chosen. We give a variable-fixing methodology that could be used in a branch-and-bound scheme based on the factorization bound for exact solution of CMESP , and we demonstrate that its ability to fix is independent of the factorization chosen. We report on successful experiments with a commercial nonlinear programming solver. We further demonstrate that the known “mixing” technique can be successfully used to combine the factorization bound with the factorization bound of the complementary CMESP and with the “linx bound” of Anstreicher. History: Andrea Lodi, area editor for Design & Analysis of Algorithms–Discrete. Funding: This work was supported by the Conselho Nacional de Desenvolvimento Científico e Tecnológico [Grants 305444/2019-0 and 434683/2018-3] and the Air Force Office of Scientific Research [Grant A9550-19-1-0175].

Programming parallel dense matrix factorizations and inversion for new-generation NUMA architectures

We propose a methodology to address the programmability issues derived from the emergence of new-generation shared-memory NUMA architectures. For this purpose, we employ dense matrix factorizations and matrix inversion (DMFI) as a use case, and we target two modern architectures (AMD Rome and Huawei Kunpeng 920) that exhibit configurable NUMA topologies. Our methodology pursues performance portability across different NUMA configurations by proposing multi-domain implementations for DMFI plus a hybrid task- and loop-level parallelization that configures multi-threaded executions to fix core-to-data binding, exploiting locality at the expense of minor code modifications. In addition, we introduce a generalization of the multi-domain implementations for DMFI that offers support for virtually any NUMA topology in present and future architectures.Our experimentation on the two target architectures for three representative dense linear algebra operations validates the proposal, reveals insights on the necessity of adapting both the codes and their execution to improve data access locality, and reports performance across architectures and inter- and intra-socket NUMA configurations competitive with state-of-the-art message-passing implementations, maintaining the ease of development usually associated with shared-memory programming.

Machine learning approaches for the prediction of serious fluid leakage from hydrocarbon wells

AbstractThe exploitation of hydrocarbon reservoirs may potentially lead to contamination of soils, shallow water resources, and greenhouse gas emissions. Fluids such as methane or CO2may in some cases migrate toward the groundwater zone and atmosphere through and along imperfectly sealed hydrocarbon wells. Field tests in hydrocarbon-producing regions are routinely conducted for detecting serious leakage to prevent environmental pollution. The challenge is that testing is costly, time-consuming, and sometimes labor-intensive. In this study, machine learning approaches were applied to predict serious leakage with uncertainty quantification for wells that have not been field tested in Alberta, Canada. An improved imputation technique was developed by Cholesky factorization of the covariance matrix between features, where missing data are imputed via conditioning of available values. The uncertainty in imputed values was quantified and incorporated into the final prediction to improve decision-making. Next, a wide range of predictive algorithms and various performance metrics were considered to achieve the most reliable classifier. However, a highly skewed distribution of field tests toward the negative class (nonserious leakage) forces predictive models to unrealistically underestimate the minority class (serious leakage). To address this issue, a combination of oversampling, undersampling, and ensemble learning was applied. By investigating all the models on never-before-seen data, an optimum classifier with minimal false negative prediction was determined. The developed methodology can be applied to identify the wells with the highest likelihood for serious fluid leakage within producing fields. This information is of key importance for optimizing field test operations to achieve economic and environmental benefits.

Development trends in suburban real estate sector

The authors of the research work, based on the analysis of market development trends and key success factors, form a SWOT matrix for out-of-town real estate objects under construction and calculate the evaluation of the matrix factors, which are used to determine the current and desired position of the object of construction in the positioning field. The main research methods are formal-logical, abstract, and empirical. Statistical methods (groupings, averages) and methods of expert evaluations were used to determine the most significant factors of SWOT matrix and to develop perspective directions of development projects' development. In order to improve market positions of out-of-town real estate objects under construction and to transfer their desirable position on the positioning field, the authors elaborate directions of development projects' development. These include the introduction of "green building" and energy-saving technologies, the use of modular construction, BIM technologies and the use of innovative materials. To increase the attractiveness of the objects under construction, development companies are recommended to carry out integrated development of the territories in the construction of cottage settlements. To increase the affordability of built objects, conversion of potential demand into real demand, the authors recommend the active use of mortgage lending system in the construction of residential housing, development and implementation of infill development of individual territories.

Real-Time Sequential Conic Optimization for Multi-Phase Rocket Landing Guidance

We introduce a multi-phase rocket landing guidance framework that can handle nonlinear dynamics and does not mandate any additional mixed-integer or nonconvex constraints to handle discrete temporal events/switching. To achieve this, we first introduce sequential conic optimization (seco), a new paradigm for solving nonconvex optimal control problems that is entirely devoid of matrix factorizations and inversions. This framework combines sequential convex programming (SCP) and first-order conic optimization and can solve unified multi-phase trajectory optimization problems in real-time. The novel features of this framework are: (1) time-interval dilation, which enables multi-phase trajectory optimization with free-transition-time; (2) single-crossing compound state-triggered constraints, which are entirely convex if the trigger and constraint conditions are convex; (3) virtual state, which is a new approach to handling artificial infeasibility in SCP methods that preserves the shapes of the constraint sets; and, (4) the use of the proportional-integral projected gradient method (pipg), a high-performance first-order conic optimization solver, in tandem with the penalized trust region (ptr) SCP algorithm. We demonstrate the efficacy and real-time capability of seco by solving a relevant multi-phase rocket landing guidance problem with nonlinear dynamics and convex constraints only, and observe that our solver is 2.7 times faster than a state-of-the-art convex optimization solver.

D-Gaussian Pell-Lucas Polynomials and Their Matrix Representations

We define a new generalization of Gaussian Pell-Lucas polynomials. We call it $d-$Gaussian Pell-Lucas polynomials. Then we present the generating function and Binet formula for the polynomials. We give a matrix representation of $d-$Gaussian Pell-Lucas polynomials. Using the Riordan method, we obtain the factorizations of Pascal matrix involving the polynomials.

Graph regularized non-negative matrix factorization with prior knowledge consistency constraint for drug–target interactions prediction

BackgroundIdentifying drug–target interactions (DTIs) plays a key role in drug development. Traditional wet experiments to identify DTIs are expensive and time consuming. Effective computational methods to predict DTIs are useful to narrow the searching scope of potential drugs and speed up the process of drug discovery. There are a variety of non-negativity matrix factorization based methods to predict DTIs, but the convergence of the algorithms used in the matrix factorization are often overlooked and the results can be further improved.ResultsIn order to predict DTIs more accurately and quickly, we propose an alternating direction algorithm to solve graph regularized non-negative matrix factorization with prior knowledge consistency constraint (ADA-GRMFC). Based on known DTIs, drug chemical structures and target sequences, ADA-GRMFC at first constructs a DTI matrix, a drug similarity matrix and a target similarity matrix. Then DTI prediction is modeled as the non-negative factorization of the DTI matrix with graph dual regularization terms and a prior knowledge consistency constraint. The graph dual regularization terms are used to integrate the information from the drug similarity matrix and the target similarity matrix, and the prior knowledge consistency constraint is used to ensure the matrix decomposition result should be consistent with the prior knowledge of known DTIs. Finally, an alternating direction algorithm is used to solve the matrix factorization. Furthermore, we prove that the algorithm can converge to a stationary point. Extensive experimental results of 10-fold cross-validation show that ADA-GRMFC has better performance than other state-of-the-art methods. In the case study, ADA-GRMFC is also used to predict the targets interacting with the drug olanzapine, and all of the 10 highest-scoring targets have been accurately predicted. In predicting drug interactions with target estrogen receptors alpha, 17 of the 20 highest-scoring drugs have been validated.

Cauchy problem for matrix factorizations of the Helmholtz equation in the space R^m

In this paper, we consider the problem of recovering solutions for matrix factorizations of the Helmholtz equation in a three-dimensional bounded domain from their values on a part of the boundary of this domain, i.e., the Cauchy problem. An approximate solution to this problem is constructed based on the Carleman matrix method.

Fault Detection and Isolation of Industrial Fermentation Process Based on Semi-supervised Convex Nonnegative Matrix Factorizations

Fault Detection and Isolation of Industrial Fermentation Process Based on Semi-supervised Convex Nonnegative Matrix Factorizations

An Index Theorem for Quarter-Plane Toeplitz Operators via Extended Symbols and Gapped Invariants Related to Corner States

In this paper, we discuss index theory for Toeplitz operators on a discrete quarter-plane of two-variable rational matrix function symbols. By using Gohberg–Kreĭn theory for matrix factorizations, we extend the symbols defined originally on a two-dimensional torus to some three-dimensional sphere and derive a formula to express their Fredholm indices through extended symbols. Variants for families of (self-adjoint) Fredholm quarter-plane Toeplitz operators and those preserving real structures are also included. For some bulk-edge gapped single-particle Hamiltonians of finite hopping range on a discrete lattice with a codimension-two right angle corner, topological invariants related to corner states are provided through extensions of bulk Hamiltonians.

Fast and Accurate Least-Mean-Squares Solvers for High Dimensional Data.

Least-mean-squares (LMS) solvers such as Linear / Ridge-Regression and SVD not only solve fundamental machine learning problems, but are also the building blocks in a variety of other methods, such as matrix factorizations. We suggest an algorithm that gets a finite set of n d-dimensional real vectors and returns a subset of d+1 vectors with positive weights whose weighted sum is exactly the same. The constructive proof in Caratheodory's Theorem computes such a subset in O(n2d2) time and thus not used in practice. Our algorithm computes this subset in O(nd+d4logn) time, using O(logn) calls to Caratheodory's construction on small but "smart" subsets. This is based on a novel paradigm of fusion between different data summarization techniques, known as sketches and coresets. For large values of d, we suggest a faster construction that takes O(nd) time and returns a weighted subset of O(d) sparsified input points. Here, a sparsified point means that some of its entries were set to zero. As an application, we show how to boost the performance of existing LMS solvers, such as those in scikit-learn library, up to x100. Generalization for streaming and distributed data is trivial. Extensive experimental results and open source code are provided.

Hybrid models for homological projective duals and noncommutative resolutions

We study hybrid models arising as homological projective duals (HPD) of certain projective embeddings $f:X\rightarrow\mathbb{P}(V)$ of Fano manifolds $X$. More precisely, the category of B-branes of such hybrid models corresponds to the HPD category of the embedding $f$. B-branes on these hybrid models can be seen as global matrix factorizations over some compact space $B$ or, equivalently, as the derived category of the sheaf of $\mathcal{A}$-modules on $B$, where $\mathcal{A}$ is an $A_{\infty}$ algebra. This latter interpretation corresponds to a noncommutative resolution of $B$. We compute explicitly the algebra $\mathcal{A}$ by several methods, for some specific class of hybrid models, and find that in general it takes the form of a smash product of an $A_{\infty}$ algebra with a cyclic group. Then we apply our results to the HPD of $f$ corresponding to a Veronese embedding of projective space and the projective embedding of Fano complete intersections in $\mathbb{P}^{n}$.

A time‐varying process monitoring approach based on iterative‐updated semi‐supervised nonnegative matrix factorizations

AbstractThis paper proposes a new time‐varying process monitoring approach based on iterative‐updated semi‐supervised nonnegative matrix factorizations (ISNMFs). ISNMFs are a type of semi‐supervised model that constructs a semi‐nonnegative matrix factorization (SNMF) model of a process using both labelled and unlabelled samples. Compared with the existing nonnegative matrix factorizations (NMFs) where NMFs are referred to as matrix factorization algorithms that factorize a nonnegative matrix into two low‐rank nonnegative matrices whose product can well approximate the original nonnegative matrix, ISNMFs have advantages in terms of the model update and the use of labelled samples. The ISNMFs‐based process monitoring approach concerns fault detection and isolation and updates an SNMF model iteratively using the latest samples to capture the change of statistical property of time‐varying processes. Moreover, the proposed fault detection and isolation approach is supported by the k‐means algorithm in theory. At last, we demonstrate the superiority of ISNMFs over the existing NMFs in terms of fault detection and isolation through a case study on the penicillin fermentation process.

CircFOXO3 protects against osteoarthritis by targeting its parental gene FOXO3 and activating PI3K/AKT-mediated autophagy

Osteoarthritis (OA) is a degenerative joint disorder causing pain and functional disability. Emerging evidence reveals that circular RNAs (circRNAs) play essential roles in OA progression and development. This study aimed to investigate the role of a novel circRNA factor, circFOXO3, in the progression of OA and elucidate its underlying molecular mechanism. The function of circFOXO3 in OA and interaction between circFOXO3 and its downstream mRNA target, forkhead box O3 (FOXO3), were evaluated by western blot (WB), immunofluorescence (IF), RNA immunoprecipitation, reverse transcription-quantitative PCR (RT-qPCR), and fluorescence in situ hybridization (FISH). Upregulation of circFOXO3 and autophagic flux were detected both in vivo and in vitro by WB, transmission electron microscopy (TEM), IF, and immunohistochemistry (IHC). A mouse model of OA was also used to confirm the role of circFOXO3 in OA pathogenesis in vivo. Decreased expression of circFOXO3 in OA cartilage tissues was directly associated with excessive apoptosis and imbalance between anabolic and catabolic factors of the extracellular matrix (ECM). Mechanistically, circFOXO3 functioned in cartilage by targeting its parental gene FOXO3 and activating autophagy. Intra-articular injection of lentivirus-circFOXO3 alleviated OA in the mouse model. In conclusion, our results reveal the key role played by circFOXO3 in OA progression; circFOXO3 overexpression may alleviate apoptosis of chondrocytes and promote anabolism of the ECM via activation of FOXO3 and autophagy, providing a potentially effective novel therapeutic strategy for OA.

Matrix-Tree Theorem of digraphs via signless Laplacians

Tutte's Matrix-Tree Theorem gives the number of spanning trees in a connected digraph by the determinant of a submatrix of the Laplacian, which is a generalization of the original Kirchhoff's Matrix-Tree Theorem associated to undirected graphs. In this paper, we obtain a new extension of Tutte's Matrix-Tree Theorem in terms of the signless Laplaicians. Moreover, we provide an elementary proof in terms of the factorization of signless Laplacian matrix into two incidence matrices. We finally generalize our result to the directed weighted graphs.