Moore-Penrose Inverse Research Articles

Infinite-time optimal feedback control of a 3-phase asynchronous motor exploiting a nonlinear finite element model

This paper presents modelling of a squirrel-cage induction motor and an optimal control method based on suboptimal control for nonlinear systems to minimise consumed energy and power losses in an induction motor drive. A coupled motor model with optimal control as a closed-loop integrated system is proposed. For modelling of the squirrel-cage asynchronous machine, a field-circuit-mechanical finite-element (FE) model is employed, in which mechanical motion is realised by a moving-mesh method and fixed mesh approach. For the control problem purpose, a surrogate induction motor model, described in a stationary rotor reference d–q frame, is applied. The optimal control is realised by a nonlinear feedback compensator method based on the state-dependent Riccati equation (SDRE) with an infinite time horizon with the surrogate model state-dependent parametrisation (SDP). To perform the control strategy, a SDRE technique with Moore–Penrose pseudoinverse is adopted. To improve the accuracy of the optimisation procedure, a finite element model was used to calculate the motor performance.

Quality Improvement Synthetic Aperture Radar (SAR) Images Using Compressive Sensing (CS) With Moore-Penrose Inverse (MPI) and Prior From Spatial Variant Apodization (SVA).

When the locations of non-zero samples are known, the Moore-Penrose inverse (MPI) can be used for the data recovery of compressive sensing (CS). First, the prior from the locations is used to shrink the measurement matrix in CS. Then the data can be recovered by using MPI with such shrinking matrix. We can also prove that the results of data recovery from the original CS and our MPI-based method are the same mathematically. Based on such finding, a novel sidelobe-reduction method for synthetic aperture radar (SAR) and Polarimetric SAR (POLSAR) images is studied. The aim of sidelobe reduction is to recover the samples within the mainlobes and suppress the ones within the sidelobes. In our study, prior from spatial variant apodization (SVA) is used to determine the locations of the mainlobes and the sidelobes, respectively. With CS, the mainlobe area can be well recovered. Samples within the sidelobe areas are also recovered using background fusion. Our method is suitable for acquired data with large sizes. The performance of the proposed algorithm is evaluated with acquired space-borne SAR and air-borne POLSAR data. In our experiments, we use the [Formula: see text] space-borne SAR data with the size of 10000 (samples) × 10000 (samples) and [Formula: see text] POLSAR data with the size of 10000 (samples) × 26000 (samples) for sidelobe suppression. Furthermore, We also verified that, our method does not affect the polarization signatures. The effectiveness for the sidelobe suppression is qualitatively examined, and results were satisfactory.

Generalized MPD and DMP Inverses

For a generalized Drazin invertible operator A and its core–EP inverse A ◯ ​​​​ d , the expression A A ◯ ​​​​ d A plays important role in the representations of well-known generalized inverses, in theory of partial orders and for decompositions of operator A. Solving certain system of operator equations leads to introduction of new generalized inverse of A A ◯ ​​​​ d A . Since this new inverse can be expressed by the generalized Moore–Penrose inverse and generalized Drazin inverse of A, it is called a generalized MPD inverse of A A ◯ ​​​​ d A . As a dual version of generalized MPD inverse, we also consider a generalized DMP inverse. The dual core inverse and core inverse of A, respectively, are special cases of the generalized MPD inverse and generalized DMP inverse. Several properties, representations and characterizations of generalized MPD and DMP inverses are established. Applying the generalized MPD and DMP inverses, we solve corresponding systems of linear equations and minimization problems. One well-known minimization problem is a particular case of our results.

Novel Steganographic Method Based on Hermitian Positive Definite Matrix and Weighted Moore–Penrose Inverses

In this paper, we describe the concept of a new data-hiding technique for steganography in RGB images where a secret message is embedded in the blue layer of specific bytes. For increasing security, bytes are chosen randomly using a random square Hermitian positive definite matrix, which is a stego-key. The proposed solution represents a very strong key since the number of variants of positive definite matrices of order 8 is huge. Implementing the proposed steganographic method consists of splitting a color image into its R, G, and B channels and implementing two segments, which take place in several phases. The first segment refers to embedding a secret message in the carrier (image or text) based on the unique absolute elements values of the Hermitian positive definite matrix. The second segment refers to extracting a hidden message based on a stego-key generated based on the Hermitian positive definite matrix elements. The objective of the data-hiding technique using a Hermitian positive definite matrix is to embed confidential or sensitive data within cover media (such as images, audio, or video) securely and imperceptibly; by doing so, the hidden data remain confidential and tamper-resistant while the cover media’s visual or auditory quality is maintained.

Comparison of Drugs Used for Prophylaxis in Hemophilia a or B with Inhibitors: A Systematic Review and Frequentist Network Meta-Analysis of Randomized Clinical Trials

Background: Frequent infusions with hemostatic agents in hemophilia patients can lead to the formation of anti-drug antibodies that can bind with clotting factors and inhibit their function. Inhibitor formation leads to worsening of hemophilia symptoms and is difficult to treat with clotting factor concentrates and is categorized as hemophilia with inhibitors. This network meta-analysis (NMA) aims to assess the efficacy of different prophylactic drugs for hemophilia A and B with inhibitors and provide a comparison among these drugs. Methods: Cochrane and PRISMA guidelines were followed to conduct this NMA. A comprehensive literature search was performed in PubMed and Embase with keywords for “hemophilia” AND “drug therapy” from the inception of data till April 2024. We screened a total of 1052 articles and 6 randomized clinical trials (RCTs, N=457) were included in this NMA.The estimation of parameters across different models was accomplished using the graph theoretical approach, which entails constructing a Moore-Penrose pseudoinverse matrix with the Netmeta package in R (version 4.3.1). NMA was registered on PROSPERO CRD42024532136. Results: In 6 RCTs (N=457), 38 patients were treated with fitusiran prophylaxis, 114 patients with concizumab prophylaxis, 147 patients with emicizumab prophylaxis, 17 patients with FEIBA NF prophylaxis, 26 patients with AICC prophylaxis, and 115 patients were only treated on demand without prophylaxis. Patients' ages ranged from 7 to 75 years. 389 (85%) patients had hemophilia A while 68 (15%) patients had hemophilia B. Emicizumab was only used in prophylaxis of patients with hemophilia A with inhibitors. The ratios of means treated annualized rate of bleeding (ARB) ranged from 0.09 (95% CI= 0.05 to 0.16, emicizumab) to 0.38 (95% CI=0.26 to 0.56, anti-inhibitor complex concentrate, AICC) for prophylaxis as compared to on-demand treatment. Emicizumab and fitusiran were significantly better than AICC prophylaxis for the ratio of mean ARB (4.22 [2.16 to 8.25] and 4.06 [1.75 to 9.45], respectively). The ratio of mean joint ARB ranged from 0.06 (95% CI= 0.03 to 0.15, emicizumab) to 0.39 (95% CI=0.25 to 0.59, AICC) for prophylaxis as compared to on-demand treatment. Emicizumab and fitusiran were significantly better than AICC prophylaxis for the ratio of mean ARB (6.41 [2.40 to 17.12] and 4.13 [1.65 to 10.33], respectively). Emicizumab and fitusiran were also significantly better than FEIBA-NF (4.3 [1.56 to 12.5] and 2.78 [1.07 to 7.20], respectively). The odds ratio of zero bleeding events ranged from 34.62 (95% CI= 4.15 to 289, fitusiran) to 7.93 (95% CI= 0.37 to 171). The odds of achieving zero bleeding events were significantly better with fitusiran, emicizumab, and concizumab as compared to on-demand treatment. Conclusion: Emicizumab, concizumab, and fitusiran were effective as prophylactic agents and were significantly better than on-demand treatment in patients with hemophilia with inhibitors. Emicizumab and fitusiran had higher efficacy as compared to other prophylactic agents in hemophilia. More large-scale randomized head-to-head comparisons are needed to confirm these results.

Computation of Outer Inverse of Tensors Based on t‐Product

ABSTRACTTensor computations play an essential role in various fields of science and engineering, including multiway data analysis. In this study, we established a few basic properties of the range and null space of a tensor by using block circulant matrices and a discrete Fourier matrix. We then discuss the outer inverse of the tensors based on ‐product with a prescribed range and kernel of third‐order tensors. We address the relation of this outer inverse with other generalized inverses, such as the Moore–Penrose inverse, group inverse, and Drazin inverse. In addition, we present a few algorithms for computing the outer inverses of the tensors. In particular, a ‐QR decomposition based algorithm was developed to compute outer inverses. It is well known that the confidentiality of information transmitted through the virtual world grows exponentially, and color image and video security have become a significant concern when communicating over the internet. As an application, a ‐QR decomposition based algorithm was demonstrated for concealing secret color images and videos.

Weighted Moore-Penrose inverses for dual matrices and its applications

Characteristics of weighted Moore-Penrose inverses for dual matrices (W-MP-D inverse) are studied in this investigation. First, we introduce the weighted compact dual singular value decomposition (WCDSVD) on the set of dual matrices. A few equivalent conditions for the existence of the W-MP-D inverse on the set of dual matrices and several explicit representations are given using WCDSVD. Finally, the simulation of standing waves (s-waves) and traveling waves (t-waves) and the application of that simulation in the t-waves identification in the brain are given.

Condition numbers for the Moore-Penrose inverse and the least squares problem involving rank-structured matrices

Perturbation theory plays a crucial role in sensitivity analysis, which is extensively used to assess the robustness of numerical techniques. To quantify the relative sensitivity of any problem, it becomes essential to investigate structured condition numbers (CNs) via componentwise perturbation theory. This paper addresses and analyses structured mixed condition number (MCN) and componentwise condition number (CCN) for the Moore-Penrose (M-P) inverse and the minimum norm least squares (MNLS) solution involving rank-structured matrices, which include the Cauchy-Vandermonde (CV) matrices and { 1 , 1 } -quasiseparable (QS) matrices. A general framework has been developed to compute the upper bounds for MCN and CCN of rank deficient parameterized matrices. This framework leads to faster computation of upper bounds of structured CNs for CV and { 1 , 1 } -QS matrices. Furthermore, comparisons of obtained upper bounds are investigated theoretically and experimentally. In addition, the structured effective CNs for the M-P inverse and the MNLS solution of { 1 , 1 } -QS matrices are presented. Numerical tests reveal the reliability of the proposed upper bounds as well as demonstrate that the structured effective CNs are computationally less expensive and can be substantially smaller compared to the unstructured CNs.

A fast method to estimate the Moore-Penrose inverse for well-determined numerical rank matrices based on the Tikhonov regularization

A fast method to estimate the Moore-Penrose inverse for well-determined numerical rank matrices based on the Tikhonov regularization

On the rank structure of the Moore-Penrose inverse of singular k-banded matrices

It is well-established that, for an n×n singular k-banded complex matrix B, the submatrices of the Moore-Penrose inverse B† of B located strictly below (resp. above) its kth superdiagonal (resp. kth subdiagonal) have a certain bounded rank s depending on n, k and rankB. In this case, B† is said to satisfy a semiseparability condition. In this paper our focus is on singular strictly k-banded complex matrices B, and we show that the Moore-Penrose inverse of such a matrix satisfies a stronger condition, called generator representability. This means that there exist two matrices of rank at most s whose parts strictly below the kth diagonal (resp. above the kth subdiagonal) coincide with the same parts of B†. When n≥3k, we prove that s is precisely the minimum rank of these two matrices. We also illustrate through examples that when n<3k those matrices may have rank less than s.

On an inequality related to the volume of a parallelepiped

The problem of establishing an upper bound for the volume of a parallelepiped is considered by utilizing an original approach involving a skew-symmetric matrix of order four (along with its Moore–Penrose inverse). It is shown that the commonly known inequality characterizing the bound can be virtually sharpened. Similarly, a sharpening is established with respect to the Cauchy–Schwarz inequality. General properties of the Moore–Penrose inverse of a skew-symmetric matrix are discussed as well.

Granformer: A granular transformer net with linear complexity

Recently, transformer models have demonstrated excellent performance across various intelligent applications owing to their ability to understand global context through self-attention mechanism. However, the extensively investigated multiplicative-based attention mechanism is inadequate for capturing relationship-based feature representations, as the dot product cannot depict the intricate semantic information between objects. Moreover, it has a great computational burden with a complexity of O(n2), due to the global feature representation capability achieved by calculating the relationship between each token within the entire feature sequence. To solve the current problem, this paper proposes a granular transformer framework with linear complexity, wherein diverse granulation functions can be employed to supersede the prevailing multiplicative relationships, and an innovative linearization methodology in the form of matrix factorization is designed to reduce the computational burden. Relying on the intricate semantics information embedded within granular structures, the capacity for feature extraction is significantly more comprehensive. Then, a novel matrix factorization methodology is developed for the linearity of granulation-based attention, accomplished by implementing separate deformable convolution sampling and using an approximate iterative algorithm based on cubic equations to calculate the Moore–Penrose inverse. The mathematical proof that our method is approximate with the complete granulation-based attention matrix is investigated in detail. Finally, the performance of Granformer, an innovative reconfiguration of plug-and-play transformer block, is evaluated on representative intelligent applications, including 3D point cloud classification, emotion recognition and sentiment analysis, and object detection. The experimental results suggest that our methodologies outperform the state-of-the-art models.

Controlling cantilevered adaptive X-ray mirrors.

Modeling the behavior of a prototype cantilevered X-ray adaptive mirror (held from one end) demonstrates its potential for use on high-performance X-ray beamlines. Similar adaptive mirrors are used on X-ray beamlines to compensate optical aberrations, control wavefronts and tune mirror focal distances at will. Controlled by 1D arrays of piezoceramic actuators, these glancing-incidence mirrors can provide nanometre-scale surface shape adjustment capabilities. However, significant engineering challenges remain for mounting them with low distortion and low environmental sensitivity. Finite-element analysis is used to predict the micron-scale full actuation surface shape from each channel and then linear modeling is applied to investigate the mirrors' ability to reach target profiles. Using either uniform or arbitrary spatial weighting, actuator voltages are optimized using a Moore-Penrose matrix inverse, or pseudoinverse, revealing a spatial dependence on the shape fitting with increasing fidelity farther from the mount.

Solvability of a system of linear equations—an approach based on the generalized inverses determined by the Penrose equations ★ ★ The paper is dedicated to Professor Götz Trenkler on the occasion of his eightieth birthday on July 14, 2023. I am sincerely thankful for over twenty years of our friendship and collaboration! Many happy returns, my Best Man!

The paper aims to play an expository role, providing a tailored introduction to the theory of matrix generalized inverses determined by the so-called Penrose equations, with the Moore–Penrose inverse as the jewel in the crown. The tailoring is made taking into account applicability of the inverses to solvability of a system of linear equations, which covers, inter alia, the least squares method. The method is introduced in a formal, though compendious way, with an intention to support researchers who want to consciously utilize it in their investigations. Additionally, the paper points out various links between the generalized inverses and theory of projectors, indicating issues which are relevant from the perspective of physics. The article can be viewed as a sequel of [O.M. Baksalary and G. Trenkler, ‘The Moore–Penrose inverse—a hundred years on a frontline of physics research,’ Eur. Phys. J. H 46, 9 (2021)], the paper prepared to celebrate the 100th anniversary of the first definition of the Moore–Penrose inverse, which shades a spotlight on the role the inverse plays in physics.

Shortest path or random walks? A framework for path weights in network meta-analysis.

Quantifying the contributions, or weights, of comparisons or single studies to the estimates in a network meta-analysis (NMA) is an active area of research. We extend this work to include the contributions of paths of evidence. We present a general framework, based on the path-design matrix, that describes the problem of finding path contributions as a linear equation. The resulting solutions may have negative coefficients. We show that two known approaches, called shortestpath and randomwalk, are special solutions of this equation, and both meet an optimization criterion, as they minimize the sum of absolute path contributions. In general, there is an infinite set of solutions, which can be identified using the generalized inverse (Moore-Penrose pseudoinverse). We consider two further special approaches. For large networks we find that shortestpath is superior with respect to run time and variability, compared to the other approaches, and is thus recommended in practice. The path-weights framework also has the potential to answer more general research questions in NMA.

A genuine extension of the Moore–Penrose inverse to dual matrices

The Moore–Penrose (MP) inverse is a genuine extension of the matrix inverse. In the last one and a half decades, in the study of approximate synthesis in kinematic, two generalizations of the MP inverse appeared for dual real matrices. One generalization, dual MP generalized inverse (DMPGI), satisfies the four MP conditions, but does not exist for uncountably many dual real matrices. Another generalization, MP dual generalized inverse (MPDGI), always exists, but does not satisfy the four MP conditions even if DMPGI exists. Recently, a new dual MP inverse (NDMPI) appeared. In this paper, based on the singular value decomposition of dual matrices given in this article, we show that the NDMPI can be regarded as a combination of the DMPGI and MPDGI. It coincides with the DMPGI when the DMPGI exists. When the DMPGI does not exist, it is an MPDGI extension of the DMPGI. It always satisfies the last three MP conditions, and minimizes the error of the first MP condition when that MP condition is impossible to be satisfied. Given a dual matrix, its NDMPI is the unique dual matrix that satisfies the extended version of the first MP condition, and the other three original MP conditions. If the original matrix is complex, its NDMPI is exactly the MP inverse of that matrix. Furthermore, NDMPI may be used to generate the minimum-norm solution to the dual least squares problem. Numerical results on kinematic analysis and dual linear regression are reported.

Weak MPD and DMP inverses

Based on a minimal rank weak Drazin inverse and minimal rank right weak Drazin inverse, our aim is to solve weakened systems of matrix equations than already existing systems for defining the MPD and DMP inverses. These new systems have unique solutions called the weak MPD and DMP inverses, which present new types of generalized inverses involving MPD, DMP, GDC, GC, MPWG and MP-m-WG inverses. Many characterizations and expressions for weak MPD and DMP inverses are established. Consequently, we obtain several new and some known properties of MPD, DMP, GDC, GC, MPWG and MP-m-WG inverses. Furthermore, the weak MPD and DMP inverses are applied in solving linear equations. As a consequence, we get a fundamental result related to application of the Moore–Penrose inverse in solving linear equation.

Extending EP matrices by means of recent generalized inverses

It is well known that a square complex matrix is called EP if it commutes with its Moore–Penrose inverse. In this paper, new classes of matrices which extend this concept are characterized. For that, we consider commutative equalities given by matrices of arbitrary index and generalized inverses recently investigated in the literature. More specifically, these classes are characterized by expressions of type AmX=XAm\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A^mX=XA^m$$\\end{document}, where X is an outer inverse of a given complex square matrix A and m is an arbitrary positive integer. The relationships between the different classes of matrices are also analyzed. Finally, a picture presents an overview of the overall studied classes.

On linear algebra of r-Hankel and r-Toeplitz matrices with geometric sequence

AbstractMatrix theory plays a crucial role in solving practical problems and performing computational operations. In particular, specific types of matrices and their linear algebraic properties are of paramount significance for these processes. In this paper, we study the properties of r-Hankel and r-Toeplitz matrices whose entries are geometric sequences, and then the determinants, inverse matrix, generalized inverse matrix (the Moore-Penrose inverse), and spectral norms of such matrices are obtained.

Adaptive predictor-set linear model: An imputation-free method for linear regression prediction on data sets with missing values.

Linear regression (LR) is vastly used in data analysis for continuous outcomes in biomedicine and epidemiology. Despite its popularity, LR is incompatible with missing data, which frequently occur in health sciences. For parameter estimation, this shortcoming is usually resolved by complete-case analysis or imputation. Both work-arounds, however, are inadequate for prediction, since they either fail to predict on incomplete records or ignore missingness-induced reduction in prediction accuracy and rely on (unrealistic) assumptions about the missing mechanism. Here, we derive adaptive predictor-set linear model (aps-lm), capable of making predictions for incomplete data without the need for imputation. It is derived by using a predictor-selection operation, the Moore-Penrose pseudoinverse, and the reduced QR decomposition. aps-lm is an LR generalization that inherently handles missing values. It is applied on a reference data set, where complete predictors and outcome are available, and yields a set of privacy-preserving parameters. In a second stage, these are shared for making predictions of the outcome on external data sets with missing entries for predictors without imputation. Moreover, aps-lm computes prediction errors that account for the pattern of missing values even under extreme missingness. We benchmark aps-lm in a simulation study. aps-lm showed greater prediction accuracy and reduced bias compared to popular imputation strategies under a wide range of scenarios including variation of sample size, goodness of fit, missing value type, and covariance structure. Finally, as a proof-of-principle, we apply aps-lm in the context of epigenetic aging clocks, linear models that predict a person's biological age from epigenetic data with promising clinicalapplications.