Moore-Penrose Inverse Research Articles

Finding influential nodes in networks using pinning control: Centrality measures confirmed with electrochemical oscillators.

The spatiotemporal organization of networks of dynamical units can break down resulting in diseases (e.g., in the brain) or large-scale malfunctions (e.g., power grid blackouts). Re-establishment of function then requires identification of the optimal intervention site from which the network behavior is most efficiently re-stabilized. Here, we consider one such scenario with a network of units with oscillatory dynamics, which can be suppressed by sufficiently strong coupling and stabilizing a single unit, i.e., pinning control. We analyze the stability of the network with hyperbolas in the control gain vs coupling strength state space and identify the most influential node (MIN) as the node that requires the weakest coupling to stabilize the network in the limit of very strong control gain. A computationally efficient method, based on the Moore-Penrose pseudoinverse of the network Laplacian matrix, was found to be efficient in identifying the MIN. In addition, we have found that in some networks, the MIN relocates when the control gain is changed, and thus, different nodes are the most influential ones for weakly and strongly coupled networks. A control theoretic measure is proposed to identify networks with unique or relocating MINs. We have identified real-world networks with relocating MINs, such as social and power grid networks. The results were confirmed in experiments with networks of chemical reactions, where oscillations in the networks were effectively suppressed through the pinning of a single reaction site determined by the computational method.

The perturbation of the Moore–Penrose inverse of quaternion tensor via the QT-product

The perturbation of the Moore–Penrose inverse of quaternion tensor via the QT-product

The Moore–Penrose inverse of tensors via the M-product

The Moore–Penrose inverse of tensors via the M-product

Learning Moore-Penrose based residuals for robust non-blind image deconvolution

This paper proposes a deep learning-based method for image restoration given an inaccurate knowledge of the degradation. We first show how the impulse response of a Wiener filter can approximate the Moore-Penrose pseudo-inverse of the blur convolution operator. The deconvolution problem is then cast as the learning of a residual in the null space of the blur kernel, which, when added to the Wiener restoration, will satisfy the image formation model. This approach is expected to make the network capable of dealing with different blurs since only residuals associated with the Wiener filter have to be learned. Artifacts caused by inaccuracies in the blur estimation and other image formation model inconsistencies are removed by a Dynamic Filter Network. The extensive experiments carried out on several synthetic and real image datasets assert the proposed method's performance and robustness and demonstrate the advantage of the proposed method over existing ones.

Compact formula for skew-symmetric system of matrix equations

In this paper, we consider skew-Hermitian solution of coupled generalized Sylvester matrix equations encompassing *-hermicity over complex field. The compact formula of the general solution of this system is presented in terms of generalized inverses when some necessary and sufficient conditions are fulfilled. An algorithm and a numerical example are provided to validate our findings. A numerical example is carried out using determinantal representations of the Moore–Penrose inverse.

Adaptive echo state network with a recursive inverse-free weight update algorithm

Echo State Network (ESN) is widely applied in sequence prediction, physics, and economics. Moore-Penrose inversion is a typical solving process of ESN. However, in the era of big data, the traditional inversion is computationally costly and the data may be irreversible. To solve this problem, a recursive inverse-free algorithm ESN model, called RIFESN, is introduced in this article. The model also introduces an adaptive activation function based on the error between target value and predicted value, which greatly improves the nonlinear mapping capability of the network. Three simulation experiments are conducted on the Mackey-Glass-17 time series, sunspots series and daily exchange rate summary of eight countries from 1990 to 2016, respectively. The results of the three simulations show that RIFESN has better prediction accuracy and operational speed, which proves that it has a good application prospect.

Diagonalization of the cross-product matrix

The paper considers diagonalization of the cross-product matrices, i.e., skew-symmetric matrices of order three. A procedure to determine a nonsingular matrix, which yields the diagonalization is indicated. Furthermore, a method to derive the inverse of a diagonalizing matrix is proposed by means of a formula for the Moore–Penrose inverse of any matrix, which is columnwise partitioned into two matrices having disjoint ranges. This rather nonstandard method to obtain the inverse of a nonsingular matrix is appealing, as it can be applied to any diagonalizing matrix, and not only of those originating from diagonalization of the cross-product matrices. The paper provides also comments and examples demonstrating applicability of the diagonalization procedure to calculate roots of a cross-product matrix.

Weighted generalized Moore–Penrose inverse

Abstract The aim of this paper is to present the weighted generalized Moore–Penrose inverse of an operator between two Hilbert spaces as an extension of the Moore–Penrose inverse and the generalized Moore–Penrose inverse defined for an operator on a Hilbert space. Basic properties, characterizations and representations of the weighted generalized Moore–Penrose inverses are established. We extend some known results and give several new results for the generalized Moore–Penrose inverse. Applying the weighted generalized Moore–Penrose inverse, the solvability of some linear equations as well as general solution forms are obtained.

A Maximally Split and Adaptive Relaxed Alternating Direction Method of Multipliers for Regularized Extreme Learning Machines

One of the significant features of extreme learning machines (ELMs) is their fast convergence. However, in the big data environment, the ELM based on the Moore–Penrose matrix inverse still suffers from excessive calculation loads. Leveraging the decomposability of the alternating direction method of multipliers (ADMM), a convex model-fitting problem can be split into a set of sub-problems which can be executed in parallel. Using a maximally splitting technique and a relaxation technique, the sub-problems can be split into multiple univariate sub-problems. On this basis, we propose an adaptive parameter selection method that automatically tunes the key algorithm parameters during training. To confirm the effectiveness of this algorithm, experiments are conducted on eight classification datasets. We have verified the effectiveness of this algorithm in terms of the number of iterations, computation time, and acceleration ratios. The results show that the method proposed by this paper can greatly improve the speed of data processing while increasing the parallelism.

Improving Newton–Schulz Method for Approximating Matrix Generalized Inverse by Using Schemes with Memory

Some iterative schemes with memory were designed for approximating the inverse of a nonsingular square complex matrix and the Moore–Penrose inverse of a singular square matrix or an arbitrary m×n complex matrix. A Kurchatov-type scheme and Steffensen’s method with memory were developed for estimating these types of inverses, improving, in the second case, the order of convergence of the Newton–Schulz scheme. The convergence and its order were studied in the four cases, and their stability was checked as discrete dynamical systems. With large matrices, some numerical examples are presented to confirm the theoretical results and to compare the results obtained with the proposed methods with those provided by other known ones.

Reverse order law and forward order law for the (b, c)-inverse

The reverse order law and the forward order law have been studied for various types of generalized inverses. The $(b,c)$-inverse is a generalization of some well known generalized inverses, such as the Moore-Penrose inverse, the Drazin inverse, the core inverse, etc. In this paper, the reverse order law for the $(b,c)$-inverse, in a unital ring, is investigated and an equivalent condition for this law to hold for the $(b,c)$-inverse is derived. Also, some known results on this topic are generalized. Furthermore, the forward order law for the $(b,c)$-inverse in a ring with a unity is introduced, for different choices of $b$ and $c$. Moreover, as corollaries of obtained results, equivalent conditions for the reverse order law and the forward order law for the inverse along an element are derived.

On EP matrices, partial isometry matrices and their numerical range

In this paper, some properties of the numerical range of some well-known generalized inverses of matrices, such as Moore-Penrose inverse, Drazin inverse, DMP inverse and CMP inverse, are investigated. In particular, an identification of EP matrices is given via the numerical range. Also, some relations between certain orders and numerical ranges of matrices that are ordered, are presented. Moreover, some results related to the sharp points of the numerical range of a partial isometry matrix are obtained. To illustrate the main results, some numerical examples are also given.

A Cubic Class of Iterative Procedures for Finding the Generalized Inverses

This article considers the iterative approach for finding the Moore–Penrose inverse of a matrix. A convergence analysis is presented under certain conditions, demonstrating that the scheme attains third-order convergence. Moreover, theoretical discussions suggest that selecting a particular parameter could further improve the convergence order. The proposed scheme defines the special cases of third-order methods for β=0,1/2, and 1/4. Various large sparse, ill-conditioned, and rectangular matrices obtained from real-life problems were included from the Matrix-Market Library to test the presented scheme. The scheme’s performance was measured on randomly generated complex and real matrices, to verify the theoretical results and demonstrate its superiority over the existing methods. Furthermore, a large number of distinct approaches derived using the proposed family were tested numerically, to determine the optimal parametric value, leading to a successful conclusion.

Moore–Penrose inverses of certain bordered matrices

Moore–Penrose inverses of certain bordered matrices

Effective operators and their variational principles for discrete electrical network problems

Using a Hilbert space framework inspired by the methods of orthogonal projections and Hodge decompositions, we study a general class of problems (called Z-problems) that arise in effective media theory, especially within the theory of composites, for defining the effective operator. A new and unified approach is developed, based on block operator methods, for obtaining solutions of the Z-problem, formulas for the effective operator in terms of the Schur complement, and associated variational principles (e.g., the Dirichlet and Thomson minimization principles) that lead to upper and lower bounds on the effective operator. In the case of finite-dimensional Hilbert spaces, this allows for a relaxation of the standard hypotheses on positivity and invertibility for the classes of operators usually considered in such problems by replacing inverses with the Moore–Penrose pseudoinverse. As we develop the theory, we show how it applies to the classical example from the theory of composites on the effective conductivity in the periodic conductivity problem in the continuum (2d and 3d) under the standard hypotheses. After that, we consider the following three important and diverse examples (increasing in complexity) of discrete electrical network problems in which our theory applies under the relaxed hypotheses. First, an operator-theoretic reformulation of the discrete Dirichlet-to-Neumann (DtN) map for an electrical network on a finite linear graph is given and used to relate the DtN map to the effective operator of an associated Z-problem. Second, we show how the classical effective conductivity of an electrical network on a finite linear graph is essentially the effective operator of an associated Z-problem. Finally, we consider electrical networks on periodic linear graphs and develop a discrete analog to the classical example of the periodic conductivity equation and effective conductivity in the continuum.

Moore–Penrose Inverse of the Signless Laplacians of Bipartite Graphs

Moore–Penrose Inverse of the Signless Laplacians of Bipartite Graphs

Singular matrices whose Moore-Penrose inverse is tridiagonal

A variety of characterizations of nonsingular matrices whose inverse is tridiagonal (irreducible or not) have been widely investigated in the literature. One well-known such characterization is stated in terms of semiseparable matrices. In this paper, we consider singular matrices A and give necessary and sufficient conditions for the Moore-Penrose inverse of A to be tridiagonal. Our approach is based on bordering techniques, as given by Bapat and Zheng (2003). In addition, we obtain necessary conditions on A analogous to the semiseparability conditions in the nonsingular case, though in the singular case they are not sufficient, as illustrated with examples. We apply our results to give an explicit description of all the 3×3 real singular matrices and 3×3 Hermitian matrices whose Moore-Penrose inverse is irreducible and tridiagonal.

Regularization operators versus regularization strategies

Abstract In this note, we shall compare two important concepts of “regularization operators” and “regularization strategies” that appear in different classical monographs. The definition of a regularization operator is related to the Moore–Penrose inverse of the operator. In general, a regularization operator is a regularization strategy. We shall show that the converse is also true under some conditions. It is interesting to note that these two systems share analogous properties.

Moore-Penrose inverse and partial orders on Hilbert space operators

In this article we explore several aspects concerning to the Moore-Penrose inverse of a bounded linear operator. On the one hand, we study monotonicity properties of the Moore-Penrose inverse with respect to the Löwner, star, minus, sharp and diamond orders. On the other hand, we analyze the validity of the reverse order law, B†A†=(AB)†, under hypothesis of operator ranges and also under hypothesis of operator orders. Finally, we study the operator B†A† as different weighted inverses of AB.

Idempotent operator and its applications in Schur complements on Hilbert C *-module

Abstract The present study proves that T T is an idempotent operator if and only if R ( I − T ∗ ) ⊕ R ( T ) = X {\mathcal{ {\mathcal R} }}\left(I-{T}^{\ast })\oplus {\mathcal{ {\mathcal R} }}\left(T)={\mathcal{X}} and ( T ∗ T ) † = ( T † ) 2 T {\left({T}^{\ast }T)}^{\dagger }={\left({T}^{\dagger })}^{2}T . Based on the equivalent conditions of an idempotent operator and related results, it is possible to obtain an explicit formula for the Moore-Penrose inverse of 2-by-2 block idempotent operator matrix. For the 2-by-2 block operator matrix, Schur complements and generalized Schur complement are well known and studied. The range inclusions of operators and idempotency of operators are used to obtain new conditions under which we can compute the Moore-Penrose inverse of Schur complements and generalized Schur complements of operators.