Rectangular Matrices Research Articles

On the theory of operator interpolation in spaces of rectangular matrixes

The problem of constructing and studying interpolation operator polynomials of an arbitrary fixed degree, defined in spaces of rectangular matrices, which would be generalisations of the corresponding interpolation formulas in the case of square matrices, is considered. Linear interpolation formulas of various structures are constructed for rectangular matrices. Matrix polynomials, with respect to which the resulting interpolation formulas are invariant, are indicated. As a generalisation of linear formulas, formulas for quadratic interpolation and interpolation by polynomials of arbitrary fixed degree in the space of rectangular matrices are constructed. Particular cases of the obtained formulas are considered: when square matrices are chosen as nodes or when the values of the interpolated function are square matrices, as well as the case when both of these conditions are satisfied. For the last variant, the possibilities of different and identical matrix orders for nodes and function values are explored. The obtained results are based on the application of some well-known provisions of the theory of matrices and the theory of interpolation of scalar functions. The presentation of the material is illustrated by a number of examples.

Construction of a Positive Valued Scalar Function of Strictly Rectangular Complex Matrices using the Framework of Spacer Matrix Components and Related Matrices

Construction of a Positive Valued Scalar Function of Strictly Rectangular Complex Matrices using the Framework of Spacer Matrix Components and Related Matrices

Estimation of Parameters in Regression Analysis Based on QR Decomposition of Rectangular Matrices by Householder Reflections

An approach to eliminate multicollinearity problems in regression analysis using QR decomposition of rectangular matrices by Householder reflection has been proposed. The reliability of this computational procedure has been proved. Povzetek: Predložen je pristup uklanjanju problema multikolinearnosti u regresijskoj analizi korištenjem QR dekompozicije pravokutnih matrica Householderovom refleksijom. Dokazana je pouzdanost ovog proračunskog postupka.

Matrix regularization for tensor fields

Abstract We propose a novel matrix regularization for tensor fields. In this regularization, tensor fields are described as rectangular matrices, and area-preserving diffeomorphisms and local rotations of the orthonormal frame are both realized as unitary similarity transformations of matrices in a unified way. We also show that the matrix commutator corresponds to the covariantized Poisson bracket for tensor fields in the large-N limit.

Inverse free reduced universum twin support vector machine for imbalanced data classification

Imbalanced datasets are prominent in real-world problems. In such problems, the data samples in one class are significantly higher than in the other classes, even though the other classes might be more important. The standard classification algorithms may classify all the data into the majority class, and this is a significant drawback of most standard learning algorithms, so imbalanced datasets need to be handled carefully. One of the traditional algorithms, twin support vector machines (TSVM), performed well on balanced data classification but poorly on imbalanced datasets classification. In order to improve the TSVM algorithm’s classification ability for imbalanced datasets, recently, driven by the universum twin support vector machine (UTSVM), a reduced universum twin support vector machine for class imbalance learning (RUTSVM) was proposed. The dual problem and finding classifiers involve matrix inverse computation, which is one of RUTSVM’s key drawbacks. In this paper, we improve the RUTSVM and propose an improved reduced universum twin support vector machine for class imbalance learning (IRUTSVM). We offer alternative Lagrangian functions to tackle the primal problems of RUTSVM in the suggested IRUTSVM approach by inserting one of the terms in the objective function into the constraints. As a result, we obtain new dual formulation for each optimization problem so that we need not compute inverse matrices neither in the training process nor in finding the classifiers. Moreover, the smaller size of the rectangular kernel matrices is used to reduce the computational time. Extensive testing is carried out on a variety of synthetic and real-world imbalanced datasets, and the findings show that the IRUTSVM algorithm outperforms the TSVM, UTSVM, and RUTSVM algorithms in terms of generalization performance.

On the theory of interpolation of functions on sets of matrices with the Hadamard multiplication

This article is devoted to the problem of interpolation of functions defined on sets of matrices with multiplication in the sense of Hadamard and is mainly an overview. It contains some known information about the Hadamard matrix multiplication and its properties. For functions defined on sets of square and rectangular matrices, various interpolation polynomials of the Lagrange type, containing both the operation of matrix multiplication in the Hadamard sense and the usual matrix product, are given. In the case of analytic functions defined on sets of square matrices with the Hadamard multiplication, some analogues of the Lagrange type trigonometric interpolation formulas are considered. Matrix analogues of splines and the Cauchy integral are given on sets of matrices with the Hadamard multiplication. Some of its applications in the theory of interpolation are considered. Theorems on the convergence of some Lagrange interpolation processes for analytic functions defined on a set of matrices with multiplication in the Hadamard sense are proved. The results obtained are based on the application of some well-known provisions of the theory of interpolation of scalar functions. Data presentation is illustrated by a number of examples.

Spectral properties of the generalized diluted Wishart ensemble

The celebrated Marčenko–Pastur law, that considers the asymptotic spectral density of random covariance matrices, has found a great number of applications in physics, biology, economics, engineering, among others. Here, using techniques from statistical mechanics of spin glasses, we derive simple formulas concerning the spectral density of generalized diluted Wishart matrices. These are defined as , where X and Y are diluted N × P rectangular matrices, whose entries correspond to the links of doubly-weighted random bipartite Poissonian graphs following the distribution , with the probability density ϱ(x, y) controlling the correlation between the matrices entries of X and Y . Our results cover several interesting cases by varying the parameters of the matrix ensemble, namely, the dilution of the graph d, the rectangularity of the matrices α = N/P, and the degree of correlation of the matrix entries via the density ϱ(x, y). Finally, we compare our findings to numerical diagonalisation showing excellent agreement.

W-weighted GDMP inverse for rectangular matrices

In this article, we introduce two new generalized inverses for rectangular matrices called $W$-weighted generalized-Drazin--Moore--Penrose (GDMP) and $W$-weighted generalized-Drazin-reflexive (GDR) inverses. The first generalized inverse can be seen as a generalization of the recently introduced GDMP inverse for a square matrix to a rectangular matrix. The second class of generalized inverse contains the class of the first generalized inverse. We then exploit their various properties and establish that the proposed generalized inverses coincide with different well-known generalized inverses under certain assumptions. We also obtain a representation of $W$-weighted GDMP inverse employing EP-core nilpotent decomposition. We define the dual of $W$-weighted GDMP inverse and obtain analogue results. Further, we discuss additive properties, reverse- and forward-order laws for GD, $W$-weighted GD, GDMP, and $W$-weighted GDMP generalized inverses.

On 2MP-, MP2- and C2MP-inverses for rectangular matrices

This paper introduces 2MP-inverses, MP2-inverses, and C2MP-inverses, for rectangular matrices following a different approach to that used in the recent literature. These new inverses generalize some classical inverses in the literature. Instead of considering a system of matrix equations as usually, in order to define 2MP-inverses and MP2-inverses, we consider a construction from oblique projectors represented by means of outer generalized inverses. We use an adequate equivalence relation, and then we pass to the quotient set in order to get the most simple canonical representative. An interesting advantage of our extension of CMP inverses from square to rectangular matrices is that we do not need any auxiliary weight matrix, but we are using the own matrix A for doing it. In addition, some properties and representations of 2MP-, MP2-, and C2MP-inverses are given.

Local Laws for Sparse Sample Covariance Matrices

We proved the local Marchenko–Pastur law for sparse sample covariance matrices that corresponded to rectangular observation matrices of order n×m with n/m→y (where y>0) and sparse probability npn>logβn (where β>0). The bounds of the distance between the empirical spectral distribution function of the sparse sample covariance matrices and the Marchenko–Pastur law distribution function that was obtained in the complex domain z∈D with Imz>v0>0 (where v0) were of order log4n/n and the domain bounds did not depend on pn while npn>logβn.

An efficient matrix iteration family for finding the generalized outer inverse

This paper presents a new iterative family for computing the generalized outer inverse with a prescribed range and null space of a given complex matrix. It is proved that the proposed methods achieve at least ninth-order of convergence. In general, the improved formulation of scheme uses only seven matrix multiplications at each iteration, but for the specific parameters, it uses only five matrix multiplications. The theoretical discussion on computational efficiency index is presented. Further, numerical results obtained are compared with existing robust methods to verify the theoretical analysis and higher computational efficiency. Several numerical examples, including rectangular, rank-deficient, large sparse, and non-singular matrices from the matrix computation tool-box (mctoolbox) are included. Further, the performance of methods is measured on randomly generated singular matrices and boundary value problem. It is demonstrated that the presented scheme gives improved results than the existing Schulz-type iterative methods for calculating the generalized inverse.

Edge statistics of large dimensional deformed rectangular matrices

We consider the edge statistics of large dimensional deformed rectangular matrices of the form Yt=Y+tX, where Y is a p×n deterministic signal matrix whose rank is comparable to n, X is a p×n random noise matrix with i.i.d. entries of mean zero and variance n−1, and t>0 gives the noise level. This model is referred to as the interference-plus-noise matrix in the study of massive multiple-input multiple-output (MIMO) system, which belongs to the category of the so-called signal-plus-noise model. For the case t=1, the spectral statistics of this model have been studied to a certain extent in the literature (Dozier and Silverstein, 2007[17,18]; Vallet et al., 2012). In this paper, we study the singular value and singular vector statistics of Yt around the right-most edge of the spectrum in the harder regime n−1/3≪t≪1. This regime is harder than the t=1 case, because on the one hand, the edge behavior of the empirical spectral distribution (ESD) of YY⊤ has a strong effect on the edge statistics of YtYt⊤ for a “small” t≪1, while on the other hand, the edge eigenvalue behavior of YtYt⊤ is not merely a perturbation of that of YY⊤ for a “large” t≫n−1/3. Under certain regularity assumptions on Y, we prove the Tracy–Widom law for the edge eigenvalues, the eigenvalue rigidity, and eigenvector delocalization for the matrices YtYt⊤ and Yt⊤Yt. These results can be used to estimate and infer the massive MIMO system. To prove the main results, we analyze the edge behavior of the asymptotic ESD of YtYt⊤ and establish optimal local laws on its resolvent. These results are of independent interest, and can be used as important inputs for many other problems regarding the spectral statistics of Yt.

Right large deviation principle for the top eigenvalue of the sum or product of invariant random matrices

In this note we study the right large deviation of the top eigenvalue (or singular value) of the sum or product of two random matrices A and B as their dimensions goes to infinity. We consider a general framework containing the cases where A and/or B are taken from an invariant ensemble or are fixed diagonal matrices. We show that the tilting method introduced in Guionnet and Maïda (2020 Electron. J. Probab. 25 1–24) can be extended to our general setting and is equivalent to the study of a spherical spin glass model specific to the operation—sum of symmetric matrices/product of symmetric matrices/sum of rectangular matrices—we are considering.

Fast commutative matrix algorithms

We show that the product of an n×3 matrix and a 3×3 matrix over a commutative ring can be computed using 6n+3 multiplications. For two 3×3 matrices this gives us an algorithm using 21 multiplications. This is an improvement with respect to Makarov's algorithm using 22 multiplications (Makarov, 1986). We generalize our result for n×3 and 3×3 matrices and present an algorithm for computing the product of an l×n matrix and an n×m matrix over a commutative ring for odd n using n(lm+l+m−1)/2 multiplications if m is odd and using (n(lm+l+m−1)+l−1)/2 multiplications if m is even. Waksman's algorithm (Waksman, 1970) and its generalization to rectangular matrices by Islam (2009) for odd n needs (n−1)(n2+2n−1)/2+n2 multiplications and (n−1)(lm+l+m−1)/2+lm multiplications respectively, thus for even and odd m less multiplications are required by our algorithm. We also give an algorithm for even n using n(lm+l+m−1)/2 multiplications without making use of divisions. Although Waksman's algorithm (Waksman, 1970) and its generalization to rectangular matrices by Islam (2009) use the same number of multiplications our algorithm is an improvement over these algorithms since they make use of some divisions by 2 and we are able to avoid them. Furthermore we present a novelty for matrix multiplication: In this paper we show that some algorithms with special properties that we refer to as weakly bilinear can be used as recursive algorithms. In comparison to bilinear algorithms for small n×n matrices say n<20 we obtain some better results. From these weakly bilinear algorithms we finally obtain approximate weakly bilinear algorithms. For instance we obtain an approximate weakly bilinear algorithm for 5×5 matrices that uses only 87 multiplications. If at all it is possible to compare this algorithm with a bilinear algorithm we obtain a better result with respect to the algorithm in Sedoglavic and Smirnov (2021).

Zeroing Neural Network for Pseudoinversion of an Arbitrary Time-Varying Matrix Based on Singular Value Decomposition

Many researchers have investigated the time-varying (TV) matrix pseudoinverse problem in recent years, for its importance in addressing TV problems in science and engineering. In this paper, the problem of calculating the inverse or pseudoinverse of an arbitrary TV real matrix is considered and addressed using the singular value decomposition (SVD) and the zeroing neural network (ZNN) approaches. Since SVD is frequently used to compute the inverse or pseudoinverse of a matrix, this research proposes a new ZNN model based on the SVD method as well as the technique of Tikhonov regularization, for solving the problem in continuous time. Numerical experiments, involving the pseudoinversion of square, rectangular, singular, and nonsingular input matrices, indicate that the proposed models are effective for solving the problem of the inversion or pseudoinversion of time varying matrices.

Constructions of rank metric codes under actions of the unitary groups

Some rank metric codes are constructed based on the actions of cyclic unitary groups and Abelian noncyclic unitary groups. All the rank metric codes studied in this paper consist of proper rectangular matrices. Most of these codes are maximum linear rank metric codes, and one is nonlinear maximum rank metric codes.

A rectangular additive convolution for polynomials

Motivated by the study of singular values of random rectangular matrices, we define and study the rectangular additive convolution of polynomials with nonnegative real roots. Our definition directly generalizes the asymmetric additive convolution introduced by Marcus, Spielman and Srivastava (2015), and our main theorem gives the corresponding generalization of the bound on the largest root from that paper. The main tool used in the analysis is a differential operator derived from the "rectangular Cauchy transform" introduced by Benaych-Georges (2009). The proof is inductive, with the base case requiring a new nonasymptotic bound on the Cauchy transform of Gegenbauer polynomials which may be of independent interest.Mathematics Subject Classifications: 26C10, 33C45

Frame soft shrinkage operators are proximity operators

In this paper, we show that the commonly used frame soft shrinkage operator, that maps a given vector x ∈ R N onto the vector T † S γ T x , is already a proximity operator, which can therefore be directly used in corresponding splitting algorithms. In our setting, the frame transform matrix T ∈ R L × N with L ≥ N has full rank N , T † denotes the Moore-Penrose inverse of T , and S γ is the usual soft shrinkage operator with threshold parameter γ > 0 . Our result generalizes the known assertion that T ⁎ S γ T is the proximity operator of ‖ T ⋅ ‖ 1 if T is an orthogonal (square) matrix. It is well-known that for rectangular frame matrices T with L > N , the proximity operator of ‖ T ⋅ ‖ 1 does not have a closed representation and needs to be computed iteratively. We show that the frame soft shrinkage operator T † S γ T is a proximity operator as well, thereby motivating its application as a replacement of the exact proximity operator of ‖ T ⋅ ‖ 1 . We further give an explanation, why the usage of the frame soft shrinkage operator still provides good results in various applications. In particular, we provide some properties of the subdifferential of the convex functional Φ which leads to the proximity operator prox Φ = T † S γ T and show that T † S γ T behaves similarly as prox ‖ T ⋅ ‖ 1 .

Restructuring of the Model “State–Probability of Choice” Based on Products of Stochastic Rectangular Matrices

Restructuring of the Model “State–Probability of Choice” Based on Products of Stochastic Rectangular Matrices

Spectroscopic properties of He4 within a multiphonon approach

Bulk and spectroscopic properties of 4He are studied within an equation of motion phonon method. Such a method generates a basis of n-phonon (n = 0, 1, 2, 3...) states composed of tensor products of particle-hole Tamm-Dancoff phonons and then solves the full eigenvalue problem in such a basis. The method does not rely on any approximation and is free of any contamination induced by the center of mass, in virtue of a procedure exploiting the singular value decomposition of rectangular matrices. Two potentials, both derived from the chiral effective field theory, are adopted in a self-consistent calculation performed within a space including up to three phonons. The latter basis states are treated under a simplifying assumption. A comparative analysis with the experimental data points out the different performances of the two potentials. It shows also that the calculation succeeds only partially in the description of the spectroscopic properties and suggests a recipe for further improvements.