Matrix Factorizations Research Articles

Zero-sum triangles for involutory, idempotent, nilpotent and unipotent matrices

In some matrix formations, factorizations and transformations, we need special matrices with some properties and we wish that such matrices should be easily and simply generated and of integers. In this paper, we propose a zero-sum rule for the recurrence relations to construct integer triangles as triangular matrices with involutory, idempotent, nilpotent and unipotent properties, especially nilpotent and unipotent matrices of index 2. With the zero-sum rule we also give the conditions for the special matrices and the generic methods for the generation of those special matrices. Some of the generated integer triangles have been found by other methods, but most of them are newly discovered, and many combinatorial identities can be found with them. The results may also interest the economists for trading analysis and simulation.

Graded matrix factorizations of size two and reduction

We associate a complete intersection singularity to a graded matrix factorization of size two of a polynomial in three variables. We show that we get an inverse to the reduction of singularities considered by C. T. C. Wall. We study this for the full strongly exceptional collections in the triangulated category of graded matrix factorizations constructed by H. Kajiura, K. Saito, and the second author.

Micro failure analysis of fiber metal laminates

In the failure analysis of composite materials, the damage of fiber and matrix is always noticed rather than the debonding of interface which is responsible for transferring the load between matrix and fiber. The stress amplification factors of fiber, matrix, and interface were extracted by establishing representative volume element model with interface and the strength of corresponding constituent is calculated in our work. Micro mechanics of failure served as the initial failure criterion of material components, and user-material subroutine (UMAT) was applied to realize progressive damage analysis of fiber metal laminate (FML) under different loading angles. The analysis results show that it is not strictly correct to directly use the stress on the matrix to calculate interface stress, and the damage trend of interface is consistent with that of fiber. The strain hardening phenomenon of FML decreases with larger loading angle, and the mechanical response of FML is about 45° symmetric due to its structure symmetry. The simulation results are in good agreement with the experiment results.

Accuracy of mutational signature software on correlated signatures

Mutational signatures are characteristic patterns of mutations generated by exogenous mutagens or by endogenous mutational processes. Mutational signatures are important for research into DNA damage and repair, aging, cancer biology, genetic toxicology, and epidemiology. Unsupervised learning can infer mutational signatures from the somatic mutations in large numbers of tumors, and separating correlated signatures is a notable challenge for this task. To investigate which methods can best meet this challenge, we assessed 18 computational methods for inferring mutational signatures on 20 synthetic data sets that incorporated varying degrees of correlated activity of two common mutational signatures. Performance varied widely, and four methods noticeably outperformed the others: hdp (based on hierarchical Dirichlet processes), SigProExtractor (based on multiple non-negative matrix factorizations over resampled data), TCSM (based on an approach used in document topic analysis), and mutSpec.NMF (also based on non-negative matrix factorization). The results underscored the complexities of mutational signature extraction, including the importance and difficulty of determining the correct number of signatures and the importance of hyperparameters. Our findings indicate directions for improvement of the software and show a need for care when interpreting results from any of these methods, including the need for assessing sensitivity of the results to input parameters.

Hochschild cohomology of Fermat type polynomials with non–abelian symmetries

For a polynomial f=x1n+…+xNn let Gf be the non–abelian maximal group of symmetries of f. This is a group generated by all g∈GL(N,C), rescaling and permuting the variables, so that f(x)=f(g⋅x). For any G⊆Gf we compute explicitly Hochschild cohomology of the category of G–equivariant matrix factorizations of f. We introduce the pairing on it showing that it is a Frobenius algebra.

Updating $ QR $ factorization technique for solution of saddle point problems

<abstract><p>We consider saddle point problem and proposed an updating $ QR $ factorization technique for its solution. In this approach, instead of working with large system which may have a number of complexities such as memory consumption and storage requirements, we computed $ QR $ factorization of matrix $ A $ and then updated its upper triangular factor $ R $ by appending the matrices $ B $ and $ \left(\begin{array}{cc} B^T & -C \\ \end{array} \right) $ to obtain the solution. The $ QR $ factorization updated process consisting of updating of the upper triangular factor $ R $ and avoid the involvement of orthogonal factor $ Q $ due to its expensive storage requirements. This technique is also suited as an updating strategy when $ QR $ factorization of matrix $ A $ is available and it is required that matrices of similar nature be added to its right end or at bottom position for solving the modified problems. Numerical tests are carried out to demonstrate the applications and accuracy of the proposed approach.</p></abstract>

Riemann–Roch for stacky matrix factorizations

AbstractWe establish a Hirzebruch–Riemann–Roch-type theorem and a Grothendieck–Riemann–Roch-type theorem for matrix factorizations on quotient Deligne–Mumford stacks. For this, we first construct a Hochschild–Kostant–Rosenberg-type isomorphism explicit enough to yield a categorical Chern character formula. Then, we find an expression of the canonical pairing of Shklyarov under the isomorphism.

Free Energy Node Embedding via Generalized Skip-gram with Negative Sampling

A widely established set of unsupervised node embedding methods can be interpreted as consisting of two distinctive steps: i) the definition of a similarity matrix based on the graph of interest followed by ii) an explicit or implicit factorization of such matrix. Inspired by this viewpoint, we propose improvements in both steps of the framework. On the one hand, we propose to encode node similarities based on the free energy distance, which interpolates between the shortest path and the commute time distances, thus, providing an additional degree of flexibility. On the other hand, we propose a matrix factorization method based on a loss function that generalizes that of the skip-gram model with negative sampling to arbitrary similarity matrices. Compared with factorizations based on the widely used <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\ell _{2}$</tex-math></inline-formula> loss, the proposed method can better preserve node pairs associated with higher similarity scores. Moreover, it can be easily implemented using advanced automatic differentiation toolkits and computed efficiently by leveraging GPU resources. Node clustering, node classification, and link prediction experiments on real-world datasets demonstrate the effectiveness of incorporating free-energy-based similarities as well as the proposed matrix factorization compared with state-of-the-art alternatives.

Jacobian-Free Poincaré-Krylov Method to Determine the Stability of Periodic Orbits of Electric Power Systems

A first application of a time domain Poincaré-Krylov approach for the study of the all-important stability of periodic solutions of nonlinear power networks is reported in this work. Whilst a Newton method solves the nonlinear algebraic equations resulting from the Poincaré map method, an iterative Krylov-subspace method based on a GMRES algorithm is applied for solving the Newton correction equations. More importantly, a QR factorization of the Hessenberg matrix involved in the GMRES least square problem is implemented using Givens rotations in order to avoid the linear growth of the computational complexity in large-scale power networks. Further, the stability of periodic solutions is determined by computing the Floquet multipliers using Ritz values and the Hessenberg matrix. Numerical tests carried out on a modified three-phase version of the IEEE 118-node system with a hydro-turbine synchronous generator and a grid-tied power converter demonstrate that speedup factors up to 8 are attainable with the Krylov-Subspace approach with respect to the standard Poincaré map method. An outstanding outcome of the comparative study is that the incorporation of QR factorization to the Hessenberg least square problem outperformed the classic periodic steady-state solvers, providing further computational savings up to 50%.

Strain Transfer and Its Influencing Factors of Matrix Encapsulated Fiber Bragg Grating Sensor

Strain Transfer and Its Influencing Factors of Matrix Encapsulated Fiber Bragg Grating Sensor

Açaí production in the municipality of São Miguel do Guamá, Pará: perspective of açaí beaters

The pulp of the açaí fruit is a staple food for a large part of the population of Pará. Nonetheless, despite the significance of açaí product for the regional economy, there is a gap in the literature on the market potential for processing this fruit. Thus, the objective of this study was to identify the challenges and opportunities of açaí beaters in the development of the activity in the municipality of São Miguel do Guamá, state of Pará, Brazil. This investigation is characterized as applied research, composed of qualitative and quantitative approaches. Data collection was based on interviews with açaí beaters and with linked production chains: middlemen, waste loaders, financial support institutions, and support institutions. The results were submitted to SWOT analysis to determine the risk factors of the internal-external matrix. As an internal risk factor, it was observed the difficulty to maintain income during the harvest period, given the decrease in the availability of raw materials and the drop in the price of the product. Externally, the risk is due to the lack of strengthening of collective organizations. It was also observed that there is a great distance between the açaí beaters and the public actors of the support institutions. The SWOT analysis was effective in identifying the activity that faces severe threats and weaknesses, internally and externally, which serves as a warning to the municipality, institutions, and actors regarding the greater attention needed to its production chain, which is evidenced by the lack of investments in this activity.

Effects of age and knee osteoarthritis on the modular control of walking: A pilot study.

BackgroundOlder adults and individuals with knee osteoarthritis (KOA) often exhibit reduced locomotor function and altered muscle activity. Identifying age- and KOA-related changes to the modular control of gait may provide insight into the neurological mechanisms underlying reduced walking performance in these populations. The purpose of this pilot study was to determine if the modular control of walking differs between younger and older adults without KOA and adults with end-stage KOA.MethodsKinematic, kinetic, and electromyography data were collected from ten younger (23.5 ± 3.1 years) and ten older (63.5 ± 3.4 years) adults without KOA and ten adults with KOA (64.0 ± 4.0 years) walking at their self-selected speed. Separate non-negative matrix factorizations of 500 bootstrapped samples determined the number of modules required to reconstruct each participant’s electromyography. One-way Analysis of Variance tests assessed the effect of group on walking speed and the number of modules. Kendall rank correlations (τb) assessed the association between the number of modules and self-selected walking speed.ResultsThe number of modules required in the younger adults (3.2 ± 0.4) was greater than in the individuals with KOA (2.3 ± 0.7; p = 0.002), though neither cohorts’ required number of modules differed significantly from the unimpaired older adults (2.7 ± 0.5; p ≥ 0.113). A significant association between module number and walking speed was observed (τb = 0.350, p = 0.021) and individuals with KOA walked significantly slower (0.095 ± 0.21 m/s) than younger adults (1.24 ± 0.15 m/s; p = 0.005). Individuals with KOA also exhibited altered module activation patterns and composition (which muscles are associated with each module) compared to unimpaired adults.ConclusionThese findings suggest aging alone may not significantly alter modular control; however, the combined effects of knee osteoarthritis and aging may together impair the modular control of gait.

A proof of a conjecture of Shklyarov

We prove a conjecture of Shklyarov concerning the relationship between K. Saito’s higher residue pairing and a certain pairing on the periodic cyclic homology of matrix factorization categories. Along the way, we give new proofs of a result of Shklyarov (Corollary 2 of [Adv. Math. 292 (2016), 181–209]) and Polishchuk–Vaintrob’s Hirzebruch–Riemann–Roch formula for matrix factorizations (Theorem 4.1.4 (i) of [Duke Math. J. 161 (2012), 1863–1926]).

Flow Balancing in Modeling of Multi-Phase Flow using Non-Conformal Finite Element Meshes

The method of balancing numerical finite element flows in modeling a process of multiphase flow using non-conformal hexagonal meshes is considered. Studies have been carried out for a simple reservoir configuration and on a more complex model of a real field of high-viscosity oil in the Tatarstan. The research results showed that the balancing method allows one to obtain a conservative solution when using non-conformal finite element meshes with sufficiently large cells. At the same time, this method is completely free of problems associated with grid orientation, even for complex models containing zones with highly variable permeability. The proposed algorithm for the adaptive choice of parameters allows to do the factorization of the SLAE matrix at sufficiently small number of time steps; therefore, the computational costs of the flow balancing procedure are an order of magnitude less than the costs associated with calculating the pressure field and phase transfer. The used non-conformal finite element meshes with an arbitrary number of docked hexagons can significantly reduce the number of degrees of freedom when modeling multiphase flows in reservoirs with much small local heterogeneity and in the presence of several perforated zones. As a result, computational costs are reduced by almost an order of magnitude, and, at the same time, the required approximation accuracy is maintained. With an increase in the scale of the model and the number of operating wells, this advantage increases even more.

Multi-resolution beta-divergence NMF for blind spectral unmixing

Many datasets are obtained as a resolution trade-off between two adversarial dimensions; for example between the frequency and the temporal resolutions for the spectrogram of an audio signal, and between the number of wavelengths and the spatial resolution for a hyper/multi-spectral image. To perform blind source separation using observations with different resolutions, a standard approach is to use coupled nonnegative matrix factorizations (NMF). As opposed to most previous works focusing on the least squares error measure, which is the β-divergence for β=2, we formulate this multi-resolution NMF problem for any β-divergence, and propose a novel algorithm based on the multiplicative updates (MU). We show on numerical experiments that the MU are able to obtain high resolutions in both dimensions on two applications: (1) blind unmixing of audio spectrograms: to the best of our knowledge, this is the first time a coupled NMF model is used in this context, and (2) the fusion of hyperspectral and multispectral images: we show that the MU compete favorably with state-of-the-art algorithms in particular in the presence of non-Gaussian noise.

Pearson equations for discrete orthogonal polynomials: I. Generalized hypergeometric functions and Toda equations

AbstractThe Cholesky factorization of the moment matrix is applied to discrete orthogonal polynomials on the homogeneous lattice. In particular, semiclassical discrete orthogonal polynomials, which are built in terms of a discrete Pearson equation, are studied. The Laguerre–Freud structure semiinfinite matrix that models the shifts by ±1 in the independent variable of the set of orthogonal polynomials is introduced. In the semiclassical case it is proven that this Laguerre–Freud matrix is banded. From the well‐known fact that moments of the semiclassical weights are logarithmic derivatives of generalized hypergeometric functions, it is shown how the contiguous relations for these hypergeometric functions translate as symmetries for the corresponding moment matrix. It is found that the 3D Nijhoff–Capel discrete Toda lattice describes the corresponding contiguous shifts for the squared norms of the orthogonal polynomials. The continuous 1D Toda equation for these semiclassical discrete orthogonal polynomials is discussed and the compatibility equations are derived. It is also shown that the Kadomtesev–Petviashvilii equation is connected to an adequate deformed semiclassical discrete weight, but in this case, the deformation does not satisfy a Pearson equation.

Colorectal cancer in Crohn's disease evaluated with genes belonging to fibroblasts of the intestinal mucosa selected by NMF

Crohn's disease (CD) is a type of chronic, inflammatory bowel disease (IBD) which affects any part of the gastrointestinal tract. This study aims to understand the mechanism which activate mucosal fibroblasts in the microenvironment of the colon in CD and colorectal carcinomas and to extract fibroblasts phenotypes via a novel framework based on non-negative factorization of matrix (NMF). The results identify a fibroblast phenotype characterized by intense pro-inflammatory activity ensured by the presence of genes belonging to the APOBEC1 family, such as APOBEC3F and APOBEC3G. These results demonstrated that there is a difference in fibroblast response in producing a pro-tumorigenic effect in CD. The different activation mechanisms could represent useful biomarkers in controlling CD development without generalizing its significance as IBD.

Analysis of Network Impacts of Frequency Containment Provided by Domestic-Scale Devices Using Matrix Factorization

This paper studies the distribution network impacts of frequency containment services derived from domestic-scale devices. Risk metrics considering the likelihood and severity of violations are proposed, given uncertainty in the location of these devices. A novel linearization approach is proposed to enable detailed simulations of large-scale networks, capturing MV–LV coupling in European-style networks of over 100 000 nodes. The approach combines a novel factorization of the power flow Jacobian matrix and an efficient linearization update step. The first of these innovations improves the scalability and practicality of the linearization, reducing the memory and computational requirements by as much as twenty times, whilst the latter reduces the median linearization error by at least 50% in all networks studied. It is demonstrated that rapid voltage change (RVC) and overvoltage constraints could limit the uptake of these devices to just a fraction of one percent of customers if the response is not managed. Sensitivity analysis demonstrates that both the likelihood and severity of constraint violations increases rapidly with the size of the devices used for frequency response.

On the minimal residual methods for solving diffusion-convection SLAEs

The objective of this research is to develop and to study iterative methods in the Krylov subspaces for solving systems of linear algebraic equations (SLAEs) with non-symmetric sparse matrices of high orders arising in the approximation of multi-dimensional boundary value problems on the unstructured grids. These methods are also relevant in many applications, including diffusion-convection equations. The considered algorithms are based on constructing ATA — orthogonal direction vectors calculated using short recursions and providing global minimization of a residual at each iteration. Methods based on the Lanczos orthogonalization, AT — preconditioned conjugate residuals algorithm, as well as the left Gauss transform for the original SLAEs are implemented. In addition, the efficiency of these iterative processes is investigated when solving algebraic preconditioned systems using an approximate factorization of the original matrix in the Eisenstat modification. The results of a set of computational experiments for various grids and values of convective coefficients are presented, which demonstrate a sufficiently high efficiency of the approaches under consideration.

PyArmadillo: a streamlined linear algebra library for Python

PyArmadillo is a linear algebra library for the Python language, with the aim of closely mirroring the programming interface of the widely used Armadillo C++ library, which in turn is deliberately similar to Matlab. PyArmadillo hence facilitates algorithm prototyping with Matlab-like syntax directly in Python, and relatively straightforward conversion of PyArmadillo-based Python code into performant Armadillo-based C++ code. The converted code can be used for purposes such as speeding up Python-based programs in conjunction with pybind11, or the integration of algorithms originally prototyped in Python into larger C++ codebases. PyArmadillo provides objects for matrices and cubes, as well as over 200 associated functions for manipulating data stored in the objects. Integer, floating point and complex numbers are supported. Various matrix factorisations are provided through integration with LAPACK, or one of its high performance drop-in replacements such as Intel MKL or OpenBLAS. PyArmadillo is open-source software, distributed under the Apache 2.0 license; it can be obtained at https://pyarma.sourceforge.io or via the Python Package Index in precompiled form.