Moore-Penrose Inverse Research Articles

Fixed-time convergence ZNN model for solving rectangular dynamic full-rank matrices inversion

The Moore–Penrose inverse of dynamic matrices has found widespread application and has garnered significant attention. The zeroing neural network (ZNN) has proven to be an effective solution for computing the Moore–Penrose inverse in dynamic matrices. This paper proposes a novel unified fixed-time ZNN (UFTZNN) model designed to achieve fixed-time convergence and solve both left and right inverse problems using a single model. Theoretical analysis of the convergence and robustness of the UFTZNN model is rigorously presented. Numerical simulations comparing the UFTZNN with existing ZNN models confirm its superiority in addressing left and right inverse problems, convergence time, and robustness. The UFTZNN model is applied to the inverse kinematic tracking problem of a six-degree-of-freedom manipulator-based photoelectric tracking system to demonstrate its potential applications and effectiveness.

Signed graphs and inverses of their incidence matrices

The Laplacian matrix L of a signed graph G may or may not be invertible. We present a combinatorial formula of the Moore-Penrose inverse of L. This is achieved by finding a combinatorial formula for the Moore-Penrose inverse of an incidence matrix of G. This work generalizes related known results on incidence and Laplacian matrices of an unsigned graph. Several examples are provided to show the usefulness of these combinatorial formulas.

Early remaining-useful-life prediction applying discrete wavelet transform combined with improved semi-empirical model for high-fidelity in battery energy storage system

The recycling of lithium-ion batteries (LIBs) from electric vehicles (EVs) for augmenting the capacity of battery energy storage systems (BESS) presents a sustainable approach to leverage investment in LIBs, mitigating economic losses. This study highlights the critical role of accurate capacity estimation and remaining-useful-life (RUL) prediction in determining optimal replacement and augmentation schedules. We introduce a semi-empirical (SE) model that integrates experimental aging data with an adaptive method and an electrochemical model to capture the degradation trends of LIBs accurately. The adoption of the Moore-Penrose pseudoinverse (Pinv) method, suited for the SE model's multi-parameter, single-output nature, significantly improves capacity estimation accuracy, achieving a maximum error below 0.2 Ah. This improved SE (ISE) model accounts for capacity trends and addresses non-stationary phenomena, including noise and signal distortion during parameter identification and capacity recovery phases.To address these challenges, we propose using discrete wavelet transform (DWT) for the decomposition of non-stationary signals, adjusting the window size to signal characteristics for enhanced noise reduction. The application of DWT to denoise α and β parameters, characterized by their non-constant periodicity and noise interference, facilitates the development of DWT combined with the ISE (DWT-ISE) aging model for early RUL prediction. Experimental validation reveals that our DWT-based approach minimizes root-mean square error (RMSE) and mean absolute error (MAE) for early RUL prediction, demonstrating the model's superiority at 300 cycles for identification and achieving the lowest prediction error at 400 cycles. Furthermore, comparison results of other models (Particle filter, and long short-term memory) reveal that our DWT-ISE aging model exhibits the best RMSE and MAE on limited data. Specifically, the RMSE and MAE values for RUL prediction, underscored as 0.032 % and 0.027 % respectively at 300 cycles, are confirmed as the highest evaluation metrics.

Identification of Wind Load Exerted on the Jacket Wind Turbines from Optimally Placed Strain Gauges Using C-Optimal Design and Mathematical Model Reduction

Wind turbine towers experience complex dynamic loads during actual operation, and these loads are difficult to accurately predict in advance, which may lead to inaccurate structural fatigue and strength assessment during the structural design phase, thereby posing safety risks to the wind turbine tower. However, online monitoring of wind loads has become possible with the development of load identification technology. Therefore, an identification method for wind load exerted on wind turbine towers was developed in this study to estimate the wind loads using structural strain, which can be used for online monitoring of wind loads. The wind loads exerted on the wind turbine tower were simplified into six equivalent concentrated forces on the topside of the tower, and the initial mathematical model for wind load identification was established based on dynamic load identification theory in the frequency domain, in which many candidate sensor locations and directions were considered. Then, the initial mathematical model was expressed as a linear system of equations. A numerical example was used to verify the accuracy and stability of the initial mathematical model for the wind load identification, and the identification results indicate that the initial mathematical model combined with the Moore–Penrose inverse algorithm can provide stable and accurate reconstruction results. However, the initial mathematical model uses too many sensors, which is not conducive to engineering applications. Therefore, D-optimal and C-optimal design methods were used to reduce the dimension of the initial mathematical model and determine the location and direction of strain gauges. The C-optimal design method adopts a direct optimisation search strategy, while the D-optimal design method adopts an indirect optimisation search strategy. Then, four numerical examples of wind load identification show that dimensionality reduction of the mathematical model leads to high accuracy, in which the C-optimal design algorithm provides more robust identification results. Moreover, the fatigue damage calculated based on the load identification wind loads closely approximates that derived from finite element simulation wind load, with a relative error within 6%. Therefore, the load identification method developed in this study offers a pragmatic solution for the accurate acquisition of the actual wind load of a wind turbine tower.

Constructions and characterizations of mixed reverse-order laws for the Moore–Penrose inverse and group inverse

Abstract This paper is concerned with constructions and characterizations of matrix equalities that involve mixed products of Moore–Penrose inverses and group inverses of two matrices. We first construct a mixed reverse-order law ( A ⁢ B ) † = B ∗ ⁢ ( A ∗ ⁢ A ⁢ B ⁢ B ∗ ) # ⁢ A ∗ {(AB)^{{\dagger}}=B^{\ast}(A^{\ast}ABB^{\ast})^{\#}A^{\ast}} , and show that this matrix equality always holds through the use of a special matrix rank equality and some matrix range operations, where A and B are two matrices of appropriate sizes, ( ⋅ ) ∗ {(\,\cdot\,)^{\ast}} , ( ⋅ ) † {(\,\cdot\,)^{{\dagger}}} and ( ⋅ ) # {(\,\cdot\,)^{\#}} mean the conjugate transpose, the Moore–Penrose inverse, and the group inverse of a matrix, respectively. We then give a diverse range of variation forms of this equality, and derive necessary and sufficient conditions for them to hold. Especially, we show an interesting fact that the two reverse-order laws ( A ⁢ B ) † = B † ⁢ A † {(AB)^{{\dagger}}=B^{{\dagger}}A^{{\dagger}}} and ( A ∗ ⁢ A ⁢ B ⁢ B ∗ ) # = ( B ⁢ B ∗ ) # ⁢ ( A ∗ ⁢ A ) # {(A^{\ast}ABB^{\ast})^{\#}=(BB^{\ast})^{\#}(A^{\ast}A)^{\#}} are equivalent.

A reconstruction method for structural stress distribution of wind turbine tower using optimised mathematical model

Wind turbine towers experience complex dynamic loads during actual operation, and these loads are difficult to accurately predict in advance, which may lead to inaccurate structural strength assessment during the structural design phase, thereby posing safety risks to the wind turbine tower. Therefore, a reconstruction method for structural stress distribution of wind turbine tower was developed based on modal expansion and linear superposition theory to estimate the full-field stress distribution of a tower, which can be used for online monitoring of structural strength. The dynamic stress distribution of the tower is assumed to be the superposition of several structural modal stress distributions, in the reconstruction method for structural stress distribution developed in this study. The modes with the greatest contribution to the wind turbine structural response were selected based on the modal effective mass, and used for the base functions of an initial mathematical model for the reconstruction of structural stress distribution of the tower, in which many candidate strain gauge locations and directions were considered. Then, the initial mathematical model was expressed as a linear system of equations. Two numerical examples were used to verify the accuracy and stability of the initial mathematical model for the reconstruction of structural stress distribution of the tower, and the reconstruction results indicate that the initial mathematical model combined with the Moore–Penrose inverse algorithm can provide stable and accurate reconstruction results. However, the initial mathematical model uses too many sensors, which is not conducive to engineering applications. Therefore, D-optimal and C-optimal design methods were used to reduce the dimensions of the initial mathematical model, optimise the system of equations, and determine the location, direction and number of strain gauges. Finally, the numerical examples of stress distribution reconstruction show that dimensionality reduction of the mathematical model leads to higher accuracy, in which the C-optimal design algorithm providing a more robust reconstruction outcome.

Qualitative, statistical, and extreme properties of spectral indices of signable pseudo-invertible graphs

In this paper, we investigate the Moore-Penrose inversion of a simple connected graph. We analyze qualitative, statistical, and extreme properties of spectral indices of signable pseudo-invertible graphs. We introduce and analyze a wide class of signable pseudo-invertible simple connected graphs. It is a generalization of the classical concept of positively integrally invertible graphs due to Godsil. We present several constructions of signable pseudo-invertible graphs. We also discuss statistical properties of various spectral indices of the class of signable pseudo-invertible graphs.

Generalized Inversion of Nonlinear Operators

Inversion of operators is a fundamental concept in data processing. Inversion of linear operators is well studied, supported by established theory. When an inverse either does not exist or is not unique, generalized inverses are used. Most notable is the Moore–Penrose inverse, widely used in physics, statistics, and various fields of engineering. This work investigates generalized inversion of nonlinear operators. We first address broadly the desired properties of generalized inverses, guided by the Moore–Penrose axioms. We define the notion for general sets and then a refinement, termed pseudo-inverse, for normed spaces. We present conditions for existence and uniqueness of a pseudo-inverse and establish theoretical results investigating its properties, such as continuity, its value for operator compositions and projection operators, and others. Analytic expressions are given for the pseudo-inverse of some well-known, non-invertible, nonlinear operators, such as hard- or soft-thresholding and ReLU. We analyze a neural layer and discuss relations to wavelet thresholding. Next, the Drazin inverse, and a relaxation, are investigated for operators with equal domain and range. We present scenarios where inversion is expressible as a linear combination of forward applications of the operator. Such scenarios arise for classes of nonlinear operators with vanishing polynomials, similar to the minimal or characteristic polynomials for matrices. Inversion using forward applications may facilitate the development of new efficient algorithms for approximating generalized inversion of complex nonlinear operators.

Emission Ghost Imaging: reconstruction with data augmentation.

Ghost Imaging enables 2D reconstruction of an object even though particles transmitted or emitted by the object of interest are detected with a single pixel detector without spatial resolution. This is possible because for the particular implementation of ghost imaging presented here, the incident beam is spatially modulated with a non-configurable attenuating mask whose orientation is varied (e.g. via transverse displacement or rotation) in the course of the ghost imaging experiment. Each orientation yields a distinct spatial pattern in the attenuated beam. In many cases, ghost imaging reconstructions can be dramatically improved by factoring the measurement matrix which consists of measured attenuated incident radiation for each of many orientations of the mask at each pixel to be reconstructed as the product of an orthonormal matrix and an upper triangular matrix provided that the number of orientations of the mask () is greater than or equal to the number of pixels () reconstructed. For the case, we present a data augmentation method that enables QR factorization of the measurement matrix. To suppress noise in the reconstruction, we determine the Moore-Penrose pseudoinverse of the measurement matrix with a truncated singular value decomposition approach. Since the resulting reconstruction is still noisy, we denoise it with the Adaptive Weights Smoothing method. In simulation experiments, our method outperforms a modification of an existing alternative orthogonalization method where rows of the measurement matrix are orthogonalized by the Gram-Schmidt method. We apply our ghost imaging methods to experimental X-ray fluorescence data acquired at Brookhaven National Laboratory.

Numerical range for weighted Moore-Penrose inverse of tensor

This article first introduces the notion of weighted singular value decomposition (WSVD) of a tensor via the Einstein product. The WSVD is then used to compute the weighted Moore-Penrose inverse of an arbitrary-order tensor. We then define the notions of weighted normal tensor for an even-order square tensor and weighted tensor norm. Finally, we apply these to study the theory of numerical range for the weighted Moore-Penrose inverse of an even-order square tensor and exploit its several properties. We also obtain a few new results in matrix setting.

Euclidean and circum-Euclidean distance matrices: characterizations and interlacing property

Motivated by the inverse formula of the distance matrix of a tree and the Moore–Penrose inverse of a circum-Euclidean distance matrix (CEDM), in this paper, we study a general real square matrix M whose Moore–Penrose inverse can be expressed as the sum of a Laplacian-like matrix L and a rank one matrix. In particular, for a symmetric hollow matrix M, under an assumption, we show that M is a Euclidean distance matrix if and only if L is positive semidefinite. Based on this, we obtain a new characterization for CEDMs involving their Moore–Penrose inverses. As an application, we show that the distance matrices of block graphs and odd-cycle-clique graphs are CEDMs. Finally, we establish an interlacing property between the eigenvalues of a Euclidean distance matrix M (including the singular case) and its associated Laplacian-like matrix L, which generalizes the interlacing property proved for the distance matrices of trees.

Inverse and Moore–Penrose inverse of conditional matrices via convolution

AbstractMoore–Penrose inverse emerges in statistics, neural networks, machine learning, applied physics, numerical analysis, tensor computations, solving systems of linear equations and in many other disciplines. Especially after the 2000s, the topic of Moore–Penrose inverse has started to attract great attention by researchers and has become a popular subject. In this paper, we investigate the Moore–Penrose inverse of the conditional matrices via convolution product formula. In order to use convolution formula effectively, we derive some useful identities by using some properties of the generalized conditional sequence. Moreover, we express the Moore–Penrose inverse of the conditional matrices in the form of block matrices. Finally, we not only present more general results compared to earlier works, but also provide many novel results using analytical techniques.

Efficient Predefined-Time Adaptive Neural Networks for Computing Time-Varying Tensor Moore-Penrose Inverse.

This article proposes predefined-time adaptive neural network (PTANN) and event-triggered PTANN (ET-PTANN) models to efficiently compute the time-varying tensor Moore-Penrose (MP) inverse. The PTANN model incorporates a novel adaptive parameter and activation function, enabling it to achieve strongly predefined-time convergence. Unlike traditional time-varying parameters that increase over time, the adaptive parameter is proportional to the error norm, thereby better allocating computational resources and improving efficiency. To further enhance efficiency, the ET-PTANN model combines an event trigger with the evolution formula, resulting in the adjustment of step size and reduction of computation frequency compared to the PTANN model. By conducting mathematical derivations, the article derives the upper bound of convergence time for the proposed neural network models and determines the minimum execution interval for the event trigger. A simulation example demonstrates that the PTANN and ET-PTANN models outperform other related neural network models in terms of computational efficiency and convergence rate. Finally, the practicality of the PTANN and ET-PTANN models is demonstrated through their application for mobile sound source localization.

Progressive Learning Model for Big Data Analysis Using Subnetwork and Moore-Penrose Inverse

Progressive Learning Model for Big Data Analysis Using Subnetwork and Moore-Penrose Inverse

A class of pseudoinverse-free greedy block nonlinear Kaczmarz methods for nonlinear systems of equations

<abstract><p>In this research, we constructed a class of nonlinear greedy average block Kaczmarz methods to solve nonlinear problems without computing the Moore-Penrose pseudoinverse of the Jacobian matrix. These kinds of methods adopt the average technique of the Gaussian Kaczmarz method and combine the greedy strategy, which greatly reduces the amount of computation. The local convergence analysis and numerical experiments of the proposed methods are given. The numerical results show the effectiveness of the proposed methods.</p></abstract>

用新激活函数加快新ZNN模型求解时变矩阵Moore-Penrose逆

用新激活函数加快新ZNN模型求解时变矩阵Moore-Penrose逆

An Adaptive Low Computational Cost Alternating Direction Method of Multiplier for RELM Large-Scale Distributed Optimization

In a class of large-scale distributed optimization, the calculation of RELM based on the Moore–Penrose inverse matrix is prohibitively expensive, which hinders the formulation of a computationally efficient optimization model. Attempting to improve the model’s convergence performance, this paper proposes a low computing cost Alternating Direction Method of Multipliers (ADMM), where the original update in ADMM is solved inexactly with approximate curvature information. Based on quasi-Newton techniques, the ADMM approach allows us to solve convex optimization with reasonable accuracy and computational effort. By introducing this algorithm into the RELM model, the model fitting problem can be decomposed into a set of subproblems that can be executed in parallel to achieve efficient classification performance. To avoid the storage of expensive Hessian for large problems, BFGS with limited memory is proposed with computational efficiency. And the optimal parameter values of the step-size search method are obtained through Wolfe line search strategy. To demonstrate the superiority of our methods, numerical experiments are conducted on eight real-world datasets. Results on problems arising in machine learning suggest that the proposed method is competitive with other similar methods, both in terms of better computational efficiency as well as accuracy.

Dynamic Moore-Penrose Inversion With Unknown Derivatives: Gradient Neural Network Approach.

Finding dynamic Moore-Penrose inverses (DMPIs) in real-time is a challenging problem due to the time-varying nature of the inverse. Traditional numerical methods for static Moore-Penrose inverse are not efficient for calculating DMPIs and are restricted by serial processing. The current state-of-the-art method for finding DMPIs is called the zeroing neural network (ZNN) method, which requires that the time derivative of the associated matrix is available all the time during the solution process. However, in practice, the time derivative of the associated dynamic matrix may not be available in a real-time manner or be subject to noises caused by differentiators. In this article, we propose a novel gradient-based neural network (GNN) method for computing DMPIs, which does not need the time derivative of the associated dynamic matrix. In particular, the neural state matrix of the proposed GNN converges to the theoretical DMPI in finite time. The finite-time convergence is kept by simply setting a large parameter when there are additive noises in the implementation of the GNN model. Simulation results demonstrate the efficacy and superiority of the proposed GNN method.

On the inverse and Moore–Penrose inverse of resistance matrix of graphs with more general matrix weights

On the inverse and Moore–Penrose inverse of resistance matrix of graphs with more general matrix weights

New Identities for Moore–Penrose Inverses of Some Operator Products and Their Reverse-order Laws

New Identities for Moore–Penrose Inverses of Some Operator Products and Their Reverse-order Laws