Matrix Factorizations Research Articles

Resolving Two-Dimensional Ambiguity in Subspace-Based Frequency Estimation for Harmonic Signals

This paper studies how to solve a matrix ambiguity to find the fundamental frequency in harmonic signals based on subspace methods. Using the eigenvectors obtained from the autocorrelation matrix of the measured signal, we obtain a model for the estimation of fundamental frequency. By a connection of an invertible matrix ambiguity, the model establishes a link between two matrices whose columns both span the same signal subspace of interest. One of the two matrices comprises the obtained eigenvectors, and the other one is an unknown Vandermonde matrix with a harmonic structure which contains the information of fundamental frequency. The special structure of the estimation model will enable us to construct a linear system so that the matrix ambiguity can be solved to construct the Vandermonde matrix. Thus, the estimate of fundamental frequency can be achieved from the entries of the Vandermonde matrix. In addition, we also discuss an LU-like factorization of the Vandermonde matrix for estimation of the fundamental frequency. Simulation results illustrate the performance of the proposed methods.

On-the-fly reduced order modeling of passive and reactive species via time-dependent manifolds

One of the principal barriers in developing accurate and tractable predictive models in turbulent flows with a large number of species is to track every species by solving a separate transport equation, which can be computationally impracticable. In this paper, we present an on-the-fly reduced order modeling of reactive as well as passive transport equations to reduce the computational cost. The presented approach seeks a low-rank decomposition of the species to three time-dependent components: (i) a set of orthonormal spatial modes, (ii) a low-rank factorization of the instantaneous species correlation matrix, and (iii) a set of orthonormal species modes, which represents a low-dimensional time-dependent manifold. Our approach bypasses the need to solve the full-dimensional species to generate high-fidelity data — as it is commonly performed in data-driven dimension reduction techniques such as the principle component analysis. Instead, the low-rank components are directly extracted from the species transport equation. The evolution equations for the three components are obtained from optimality conditions of a variational principle. The time-dependence of the three components enables an on-the-fly adaptation of the low-rank decomposition to transient changes in the species. Several demonstration cases of reduced order modeling of passive and reactive transport equations are presented.

Parallel finite element solver for multi-core computers with shared memory

A direct finite element solver is considered for symmetric sparse matrices arising from the application of the finite element method to problems of structural mechanics and solid mechanics. The solver is based on the block Cholesky method generalized to indefinite matrices. A distinctive feature of the proposed method from other known approaches is the original algorithm for parallelizing the factorization procedure, as well as assembling the matrix of a system of linear and linearized algebraic equations, based on the analysis of the adjacency graph for the finite elements of the design model. All the proposed parallelization algorithms use dynamic mapping of computational tasks to threads and are based on a general idea that uses a dependency vector that controls the execution of computational tasks in a parallel region. This approach is relatively simple and at the same time demonstrates high efficiency. It was developed for solving nonlinear static and dynamic problems of structural mechanics, requiring multiple assembly and factorization of a sparse matrix, but can be successfully applied in other areas of computational mathematics.

Regularization of the Ill-Posed Cauchy Problem for Matrix Factorizations of the Helmholtz Equation on the Plane

In this paper, we present an explicit formula for the approximate solution of the Cauchy problem for the matrix factorizations of the Helmholtz equation in a bounded domain on the plane. Our formula for an approximate solution also includes the construction of a family of fundamental solutions for the Helmholtz operator on the plane. This family is parameterized by function K(w) which depends on the space dimension. In this paper, based on the results of previous works, the better results can be obtained by choosing the function K(w).

Noncommutative Homological Mirror Functor

We formulate a constructive theory of noncommutative Landau-Ginzburg models mirror to symplectic manifolds based on Lagrangian Floer theory. The construction comes with a natural functor from the Fukaya category to the category of matrix factorizations of the constructed Landau-Ginzburg model. As applications, it is applied to elliptic orbifolds, punctured Riemann surfaces and certain non-compact Calabi-Yau threefolds to construct their mirrors and functors. In particular it recovers and strengthens several interesting results of Etingof-Ginzburg, Bocklandt and Smith, and gives a unified understanding of their results in terms of mirror symmetry and symplectic geometry. As an interesting application, we construct an explicit global deformation quantization of an affine del Pezzo surface as a noncommutative mirror to an elliptic orbifold.

Fast and scalable computations for Gaussian hierarchical models with intrinsic conditional autoregressive spatial random effects

Fast algorithms are developed for Bayesian analysis of Gaussian hierarchical models with intrinsic conditional autoregressive (ICAR) spatial random effects. To achieve computational speed-ups, first a result is proved on the equivalence between the use of an improper CAR prior with centering on the fly and the use of a sum-zero constrained ICAR prior. This equivalence result then provides the key insight for the algorithms, which are based on rewriting the hierarchical model in the spectral domain. The two novel algorithms are the Spectral Gibbs Sampler (SGS) and the Spectral Posterior Maximizer (SPM). Both algorithms are based on one single matrix spectral decomposition computation. After this computation, the SGS and SPM algorithms scale linearly with the sample size. The SGS algorithm is preferable for smaller sample sizes, whereas the SPM algorithm is preferable for sample sizes large enough for asymptotic calculations to provide good approximations. Because the matrix spectral decomposition needs to be computed only once, the SPM algorithm has computational advantages over algorithms based on sparse matrix factorizations (which need to be computed for each value of the random effects variance parameter) in situations when many models need to be fitted. Three simulation studies are performed: the first simulation study shows improved performance in computational speed in estimation of the SGS algorithm compared to an algorithm that uses the spectral decomposition of the precision matrix; the second simulation study shows that for model selection computations with 10 regressors and sample sizes varying from 49 to 3600, when compared to the current fastest state-of-the-art algorithm implemented in the R package INLA, SPM computations are 550 to 1825 times faster; the third simulation study shows that, when compared to default INLA settings, SGS and SPM combined with reference priors provide much more adequate uncertainty quantification. Finally, the application of the novel SGS and SPM algorithms is illustrated with a spatial regression study of county-level median household income for 3108 counties in the contiguous United States in 2017.

Categorical Saito theory, II: Landau-Ginzburg orbifolds

Let W∈C[x1,⋯,xN] be an invertible polynomial with an isolated singularity at origin, and let G⊂SLN∩(C⁎)N be a finite diagonal and special linear symmetry group of W. In this paper, we use the category MFG(W) of G-equivariant matrix factorizations and its associated VSHS to construct a G-equivariant version of Saito's theory of primitive forms. We prove there exists a canonical categorical primitive form of MFG(W) characterized by GWmax-equivariance. Conjecturally, this G-equivariant Saito theory is equivalent to the genus zero part of the FJRW theory under LG/LG mirror symmetry. In the marginal deformation direction, we verify this for the FJRW theory of (15(x15+⋯+x55),Z/5Z) with its mirror dual B-model Landau-Ginzburg orbifold (15(x15+⋯+x55),(Z/5Z)4). In the case of the Quintic family W=15(x15+⋯+x55)−ψx1x2x3x4x5, we also prove a comparison result of B-model VSHS's conjectured by Ganatra-Perutz-Sheridan [14].

On quartic double fivefolds and the matrix factorizations of exceptional quaternionic representations

On étudie les quartiques doubles de dimension cinq du point de vue des variétés Fano de type Calabi–Yau et des représentations quaternioniques exceptionnelles. On démontre tout d’abord qu’une quartique double de dimension cinq générique peut être représentée comme un recouvrement double de ℙ 5 ramifié le long d’une section linéaire de la quartique Spin 12 -invariante dans ℙ 31 . Le nombre de telles représentations est fini. Ensuite, en utilisant la géométrie des décompositions de Vinberg de type II de certaines représentations quaternioniques exceptionnelles et en se basant sur des calculs effectués grâce au logiciel Macaulay2, on démontre l’existence d’un fibré vectoriel sphérique de rang 6 sur de telles quartiques doubles. On déduit finalement de l’existence de ces fibrés que l’unité homologique des catégories Calabi–Yau de dimension trois associées par Kuznetsov aux quartiques doubles de dimension cinq est ℂ⊕ℂ[3].

Linear algebra of the Lucas matrix

In this paper, we give the factorizations of the Lucas and inverse Lucas matrices. We also investigate the Cholesky factorization of the symmetric Lucas matrix. Moreover, we obtain the upper and lower bounds for the eigenvalues of the symmetric Lucas matrix by using some majorization techniques.

A thermal form factor series for the longitudinal two-point function of the Heisenberg–Ising chain in the antiferromagnetic massive regime

We consider the longitudinal dynamical two-point function of the XXZ quantum spin chain in the antiferromagnetic massive regime. It has a series representation based on the form factors of the quantum transfer matrix of the model. The nth summand of the series is a multiple integral accounting for all n-particle–n-hole excitations of the quantum transfer matrix. In previous works, the expressions for the form factor amplitudes appearing under the integrals were either again represented as multiple integrals or in terms of Fredholm determinants. Here, we obtain a representation which reduces, in the zero-temperature limit, essentially to a product of two determinants of finite matrices whose entries are known special functions. This will facilitate the further analysis of the correlation function.

Fusion of interfaces in Landau-Ginzburg models: a functorial approach

We investigate the fusion of B-type interfaces in two-dimensional supersymmetric Landau-Ginzburg models. In particular, we propose to describe the fusion of an interface in terms of a fusion functor that acts on the category of modules of the underlying polynomial rings of chiral superfields. This uplift of a functor on the category of matrix factorisations simplifies the actual computation of interface fusion. Besides a brief discussion of minimal models, we illustrate the power of this approach in the SU(3)/U(2) Kazama-Suzuki model where we find fusion functors for a set of elementary topological defects from which all rational B-type topological defects can be generated.

Complete coherence of random, nonstationary electromagnetic fields

Despite a wide range of applications, the coherence theory of random, nonstationary (pulsed or otherwise) electromagnetic fields is far from complete. In this work, we show that full coherence of a nonstationary vectorial field at a pair of spatiospectral points is equivalent to the factorization of the cross-spectral density matrix, and full pointwise coherence over a spatial volume and spectral band leads to a factored cross-spectral density throughout the domain. We further show that in the latter case, the time-domain mutual coherence matrix factors in the spatiotemporal variables, and the field is temporally fully coherent throughout the volume. The results of this work justify that certain expressions of random pulsed electromagnetic beams appearing in the literature can be called coherent-mode representations.

Integer matrix factorisations, superalgebras and the quadratic form obstruction

We identify and analyse obstructions to factorisation of integer matrices into products NTN or N2 of matrices with rational or integer entries. The obstructions arise as quadratic forms with integer coefficients and raise the question of the discrete range of such forms. They are obtained by considering matrix decompositions over a superalgebra. We further obtain a formula for the determinant of a square matrix in terms of adjugates of these matrix decompositions, as well as identifying a co-Latin symmetry space.

Categorical Saito theory, I: A comparison result

In this paper, we present an explicit cyclic minimal A∞ model for the category of matrix factorizations MF(W) of an isolated hypersurface singularity. The key observation is to use Kontsevich's deformation quantization technique. Pushing this idea further, we use the Tsygan formality map to obtain a comparison theorem that the categorical Variation of Semi-infinite Hodge Structure of MF(W) is isomorphic to Saito's original geometric construction in primitive form theory.

Spike-and-slab Lasso biclustering

Biclustering methods simultaneously group samples and their associated features. In this way, biclustering methods differ from traditional clustering methods, which utilize the entire set of features to distinguish groups of samples. Motivating applications for biclustering include genomics data, where the goal is to cluster patients or samples by their gene expression profiles; and recommender systems, which seek to group customers based on their product preferences. Biclusters of interest often manifest as rank-1 submatrices of the data matrix. This submatrix detection problem can be viewed as a factor analysis problem in which both the factors and loadings are sparse. In this paper, we propose a new biclustering method called Spike-and-Slab Lasso Biclustering (SSLB) which utilizes the Spike-and-Slab Lasso of Ročková and George (J. Amer. Statist. Assoc. 113 (2018) 431–444) to find such a sparse factorization of the data matrix. SSLB also incorporates an Indian Buffet Process prior to automatically choose the number of biclusters. Many biclustering methods make assumptions about the size of the latent biclusters; either assuming that the biclusters are all of the same size, or that the biclusters are very large or very small. In contrast, SSLB can adapt to find biclusters which have a continuum of sizes. SSLB is implemented via a fast EM algorithm with a variational step. In a variety of simulation settings, SSLB outperforms other biclustering methods. We apply SSLB to both a microarray dataset and a single-cell RNA-sequencing dataset and highlight that SSLB can recover biologically meaningful structures in the data. The SSLB software is available as an R/C++ package at https://github.com/gemoran/SSLB.

Least-squares reverse time migration method using the factorization of the Hessian matrix

Least-squares reverse time migration method using the factorization of the Hessian matrix

The grain refinement of aluminium by a new as-cast Al–Ti–TiCN composite

A new type of composite refiner Ti/TiCN is prepared by loading submicron-TiCN particles onto Ti micron-particles and added to industrial pure aluminium. Refining mechanism has been characterised in detail by optical microscopy, scanning electron microscopy and transmission electron microscopy coupled with energy disperse X-ray spectra. These experimental results show that Ti/TiCN inoculation is more effective in refining pure aluminium than Ti and TiCN added separately. The relationship between grain size and growth restriction factors of aluminium matrix containing Ti/TiCN refiner is d = 70.2 + 1190.8/Q. According to the free growth theory, aluminium matrix with Ti/TiCN inoculation has the largest solidification driving force to promote grain nucleation under the same growth restriction factors (Q = 0.5,1, 5, 12, 20 K) compared to Ti and TiCN refiners.

Topological defects and SUSY RG flow

We study the effect of bulk perturbations of N=(2) superconformal minimal models on topological defects. In particular, symmetries and more general topological defects which survive the flow to the IR are identified. Our method is to consider the topological subsector and make use of the Landau-Ginzburg formulation to describe RG flows and topological defects in terms of matrix factorizations.

An identification method of cashmere and wool by the two features fusion

PurposeThe purpose of this paper is to select the optimal feature parameters to further improve the identification accuracy of cashmere and wool.Design/methodology/approachTo increase the accuracy, the authors put forward a method selecting optimal parameters based on the fusion of morphological feature and texture feature. The first step is to acquire the fiber diameter measured by the central axis algorithm. The second step is to acquire the optimal texture feature parameters. This step is mainly achieved by using the variance of secondary statistics of these two texture features to get four statistics and then finding the impact factors of gray level co-occurrence matrix relying on the relationship between the secondary statistic values and the pixel pitch. Finally, the five-dimensional feature vectors extracted from the sample image are fed into the fisher classifier.FindingsThe improvement of identification accuracy can be achieved by determining the optimal feature parameters and fusing two texture features. The average identification accuracy is 96.713% in this paper, which is very helpful to improve the efficiency of detector in the textile industry.Originality/valueIn this paper, a novel identification method which extracts the optimal feature parameter is proposed.

Jacobsthal Matrices and their Properties

Objectives: Matrices with Jacobsthal numbers are used in the medical image processing applications. The Cholesky factorization of the matrix with the Jacobsthal number is anlayzed. We also investigate the upper and lower bounds of the eigenvalues of the symmetric Jacobsthal and Jacobsthal-Lucas matrices. Methods: In this paper, we define a factor matrix and use the factorization techniques to get Cholesky decomposition of the Jacobsthal, Jacobsthal-Lucas matrix and inverses of these matrices. The bounds for eigenvalues are obtained using majorization techniques. Findings: The Cholesky factorization has been obtained using the factor matrix technique for any matrix of order n with entries from the Jacobsthal and Jacobsthal Lucas sequences. Novelty: Factorization of Lucas and symmetric Lucas matrix has already been obtained using the factorization technique. In this paper we give the factorization of the matrices with entries from the Jacobsthal and Jacobsthal Lucas sequences. Mathematics Subject Classification (2020). 15A23, 11B39, 15A18,15A42 Keywords: Jacobsthal matrix; Jacobsthal-Lucas matrix; symmetric; eigenvalues