Nonsingular Research Articles

Data-adaptive binary classifiers in high dimensions using random partitioning

Classification in high dimensions has been highlighted for the past two decades since Fisher's linear discriminant analysis (LDA) is not optimal in a smaller sample size n comparing the number of covariates p, i.e. p>n, which is mostly due to the singularity of the sample covariance matrix. Rather than modifying how to estimate the sample covariance and sample mean vector in constructing a classifier, we build two types of high-dimensional classifiers using data splitting, i.e. single data splitting (SDS) and multiple data splitting (MDS). Moreover, we introduce a weighted version of MDS classifier that improves classification performance as illustrated in numerical studies. Each of the split data sets has a smaller size of covariates compared to the sample size so that LDA is applicable, and classification results can be combined with respect to minimizing the misclassification rate. We present theoretical justification backing up our proposed methods by comparing misclassification rates with LDA in high dimension. We also conduct a wide range of simulations and analyse four microarray data sets, which demonstrates that our proposed methods outperform some existing methods or at least yield comparable performances.

Quantum circuit syntheses of Boolean matrix multiplication using only CNOT gates

Abstract Boolean matrices and operations on Boolean matrices have become increasingly popular in quantum computing and quantum information because they can be used to represent quantum states and the corresponding operations. In this paper, we first give a quantum circuit direct synthesis of general Boolean matrix multiplication using Toffoli gates. We assign a qubit to each element of the matrix and add Toffoli gates at the corresponding positions where the matrix elements are multiplied according to the arithmetic of matrix multiplication. For $n\times p$ dimensional matrix $A$ multiplied by $p\times m$ dimensional matrix $B$, $np + pm + nm$ qubits and at most $pmn$ Toffoli gates are used in total. Then, an improved heuristic algorithm based on matrix Hamming distance selection of CNOT gate is proposed to synthesize the quantum circuit of a matrix, without using wire switching and SWAP gates, using only CNOT gates. The results show that our method is as good as or better than the classical synthesis. Finally, the quantum circuit of Boolean matrix multiplication with at least one invertible or full-rank matrix using only CNOT gates is given, which is divided into five cases for discussion and analysis. For $n\times n$ dimensional invertible matrix $A$ multiplied by $n\times m$ dimensional matrix $B$, $nm$ qubits and at most $mn^{2}$ CNOT gates are used in total. For $n\times p$ dimensional column full-rank matrix $A$ multiplied by $p\times m$ dimensional matrix $B$, $nm$ qubits and at most $pmn$ CNOT gates are used in total. The results show that this method uses less quantum resources to synthesize the quantum circuit of Boolean matrix multiplication. The correctness of the quantum circuits was verified by the Aer simulator of the IBM Quantum platform.

A Static/Dynamic Coupled Harmonic Balance Method for Dry Friction Systems Containing Rigid Body Modes

Abstract A harmonic balance method for under-constrained dry friction systems containing rigid body modes (HBM-RBM) is proposed. This method aims to overcome the encountered obstacle when applying the harmonic balance method to turbine blades damped by underplatform dampers. The inspiration for HBM-RBM comes from the free interface modal synthesis method. The key innovation involves deriving the elastic inversion of the singular stiffness matrix through the elimination of rigid body modes. In this way, the general HBM framework can be adopted, and the frequency response of under-constrained dry friction systems can be solved in a static/dynamic coupled manner. The accuracy and efficiency are both verified on a lumped parameter model and a finite element model of turbines with UPDs from a real gas turbine. A comparative study between the HBM-RBM and the commonly adopted way of imposing artificial grounding springs (HBM-AGS) is conducted. Results demonstrate that the HBM-RBM holds a significant advantage over HBM-AGS, as it eliminates the need for artificial grounding springs (AGS) and avoids the necessity for numerous trial cases to determine AGS stiffness.

On affine spaces of rectangular matrices with constant rank

Let F be a field, and n≥p≥r>0 be integers. In a recent article, Rubei has determined, when F is the field of real numbers, the greatest possible dimension for an affine subspace of n–by–p matrices with entries in F in which all the elements have rank r. In this note, we generalize her result to an arbitrary field with more than r+1 elements, and we classify the spaces that reach the maximal dimension as a function of the classification of the affine subspaces of invertible matrices of Ms(F) with dimension (s2). The latter is known to be connected to the classification of nonisotropic quadratic forms over F up to congruence.

Quantum Codes as an Application of Constacyclic Codes

The main focus of this paper is to analyze the algebraic structure of constacyclic codes over the ring R=Fp+w1Fp+w2Fp+w22Fp+w1w2Fp+w1w22Fp, where w12−α2=0, w1w2=w2w1, w23−β2w2=0, and α,β∈Fp∖{0}, for a prime p. We begin by introducing a Gray map defined over R, which is associated with an invertible matrix. We demonstrate its advantages over the canonical Gray map through some examples. Finally, we create new and improved quantum codes from constacyclic codes over R using Calderbank–Shore–Steane (CSS) construction.

Adaptive visual servoing control for uncalibrated robot manipulator with uncertain dead-zone constraint

This paper addresses the adaptive visual tracking control problem of an uncalibrated camera robot system with unknown dead-zone inputs. The uncertainties include camera extrinsic and intrinsic parameters, robot dynamic parameters, and feature depth parameters. The control achieves a better performance by establishing a smooth inverse model of the dead-zone input, which eliminates the nonlinear effect of the actuator constraint. A novel adaptive control scheme is presented which does not require a priori knowledge of the parameter intervals of dead-zone model, to update the parameter values online. Furthermore, adaptive laws are provided to estimate the uncertainties that exist both in robot and in camera models. Meanwhile, in order to avoid the singularity of the image Jacobian matrix, a projection algorithm is also introduced. Subsequently, a novel robust adaptive controller together with dead-zone compensation is established so that the tracking error image space converges to the origin. Simulations and experimental studies are conducted to verify the effectiveness of the proposed method.

Assessing the value of combined use of distributed acoustic sensing (DAS) and multicomponent vertical seismic profiling (VSP) data with network-based full waveform inversion and uncertainty quantification: A case study in Alberta, Canada

Distributed acoustic sensing (DAS) technology, deployed in a vertical seismic profiling (VSP) experimental configuration, has emerged as a candidate for non-disruptive and low-cost seismic monitoring of CO2 geostorage and plume evolution. Full waveform inversion (FWI) has likewise received significant attention because it uses relatively complete physical models of wave propagation and because of its sample-by-sample incorporation of data information. Recent artificial neural network-based FWI algorithms (built with, for instance, recursive neural networks, or RNNs) have added to FWI a range of flexible and efficient tools for gradient computation and options for uncertainty assessment and initial model proxies. An important current research area for the use of DAS data is to better understand how they change our confidence levels in models selected through FWI. In particular, we seek to understand whether either DAS data or conventional geophone data alone are optimal for FWI model selection in the CO2 problem, and if not, to what degree they complement each other. The Snowflake 4D VSP dataset, which includes multi-offset and multi-azimuth broadband sources illuminating both fiberoptic cable and densely sampled accelerometer phones in the borehole, was acquired by our group to directly address these questions. In this study, we quantify uncertainty (UQ) by sampling the posterior model covariance matrix from the inverse Hessian matrix at the end of RNN-FWI runs on the Snowflake baseline data, invoking a velocity-density parameterization, and involving mixtures of accelerometer and DAS data. In this UQ context, the complementary effect of combining accelerometer and DAS data is evident in the Vp and Vs models. Our confidence in the approach is bolstered by its independent prediction of a region of known high uncertainty. In the overall pursuit of reliable and low-cost monitoring tools, this supports continued consideration of a multicomponent sensors supported by DAS approach.

Enumeration of matrices with single unit-entry rows over finite commutative rings

Let [Formula: see text] be a finite commutative ring with identity. In this paper, formal expressions of the number of [Formula: see text] matrices over [Formula: see text] of rank [Formula: see text] and the number of invertible matrices over [Formula: see text] are presented. The number of matrices over [Formula: see text] with a given rank and a given number of single unit-entry rows, rows in which a single entry is a unit and all other entries are zero, is finally determined.

Effect of discretization on off-axis listeners in binaural synthesis based on the optimal source distribution

3D sound reproduction based on binaural synthesis over loudspeakers is one of the most important techniques in immersive spatial audio. The system inversion involved in the binaural synthesis gives rise to several problems such as dynamic range loss, distortion of the reproduction transfer function, and a lack of robustness to room reflections. The principle of Optimal Source Distribution was proposed to overcome these problems. This utilizes a pair of conceptual monopole transducers whose azimuthal location varies continuously as a function of frequency. Another potential of the principle is that it can provide listening positions for multiple off-axis listeners, in addition to the on-axis listener whose inverse matrix is given, since the phase interference state is aligned at all frequencies. However, these continuously varying transducers are too conceptual to realize in practice. It is necessary to provide discretized sound sources with appropriate frequency ranges according to the principle. In this paper, the effect of discretization of the principle on off-axis listeners' performance is numerically investigated to reveal how those off-axis sweet spots are formed as the discretization becomes finer. An experiment is also carried out using an actual discretized system and the effect of the head-related transfer functions is considered.

A New Galerkin Quadrature Method Not Requiring a Matrix Inverse

We derive a new Galerkin quadrature (GQ) method for S n calculations that differs from the two methods preceding it in that a matrix inverse for an N d × N d matrix, where N d is the number of directions in the quadrature set, is no longer required. Galerkin quadrature methods are designed for calculations with highly anisotropic scattering. Such methods are not simply special angular quadratures but also are methods for representing the S n scattering source that offers several advantages relative to the standard scattering source representation when highly truncated Legendre cross-section expansions must be used. Galerkin quadrature methods are also useful when the scattering is moderately anisotropic, but the quadrature being used is not sufficiently accurate for the order of the scattering source expansion that is required. We derive the new method and present computational results showing that its performance for two challenging problems is comparable to those of the two GQ methods that preceded it.

Characterizing Jordan embeddings between block upper-triangular subalgebras via preserving properties

Let Mn be the algebra of n×n complex matrices. We consider arbitrary subalgebras A of Mn which contain the algebra of all upper-triangular matrices, and their Jordan embeddings. We first describe Jordan embeddings ϕ:A→Mn as maps of the formϕ(X)=TXT−1orϕ(X)=TXtT−1, where T∈Mn is an invertible matrix, and then we obtain a simple criterion of when one block upper-triangular subalgebra Jordan-embeds into another (and in that case we describe the form of such embeddings). As a main result, we characterize Jordan embeddings ϕ:A→Mn (when n≥3) as continuous injective maps which preserve commutativity and spectrum. We show by counterexamples that all these assumptions are indispensable (unless A=Mn when injectivity is superfluous).

Fault diagnosis based on incomplete sensor variables with a hierarchical semi-supervised Gaussian mixture classifier

We propose a hierarchical semi-supervised variational semi-Bayesian Gaussian mixture classifier based on the partially incomplete and unlabeled samples for the fault diagnosis of mechanical and electrical systems. These systems are typically complex in structure and are monitored by multiple sensors simultaneously. Some sensor variables in the collected data may be incomplete due to sensor malfunctions or transmission errors. Additionally, labeling the data can be a time-consuming and labor-intensive task, resulting in many unlabeled samples. To address these challenges, the missing sensor variables and the unavailable labels are treated as hidden values and handled within the framework of the Expectation Maximization algorithm. We employ a semi-Bayesian technique with variational inference to estimate the model parameters. Specifically, we introduce prior distribution to the mean and covariance matrix to address the possible singularity of the empirical covariance matrix. The weighting coefficients are left without putting prior distribution so that their values can decay to zero, effectively removing redundant components of the mixture model. The factorized distribution is utilized to approximate the posterior distribution of the model parameters, as well as the missing labels and other latent variables. Numerical and real case studies are carried out to verify the effectiveness of the proposed technique.

An effective criterion for a stable factorisation of strictly non-singular 2 × 2 matrix functions: use of the ExactMPF package

In this paper, we propose a method to factorise arbitrary strictly non-singular 2 × 2 matrix functions, allowing for stable factorisation. For this purpose, we use the E x a c t M P F package working within the Maple environment previously developed by the authors and perform an exact factorisation of a non-singular polynomial matrix function. A crucial point in the present analysis is the evaluation of a stability region of the canonical factorisation of the polynomial matrix functions. This, in turn, allows us to propose a sufficient condition for the given matrix function admitting stable factorisation.

Strong B-T inverse

New-typed matrix inverses based on the Hartwig-Spindelböck decomposition were investigated, which is called the strong B-T inverse, as a generalization of the B-T inverse. The relationships between the above inverse and other matrix inverses were established. Several sufficient and necessary conditions of the strong B-T inverse are obtained via the column and null spaces.

Distribution of centrality measures on undirected random networks via the cavity method

The Katz centrality of a node in a complex network is a measure of the node's importance as far as the flow of information across the network is concerned. For ensembles of locally tree-like undirected random graphs, this observable is a random variable. Its full probability distribution is of interest but difficult to handle analytically because of its "global" character and its definition in terms of a matrix inverse. Leveraging a fast Gaussian Belief Propagation-Cavity algorithm to solve linear systems on tree-like structures, we show that i) the Katz centrality of a single instance can be computed recursively in a very fast way, and ii) the probability [Formula: see text] that a random node in the ensemble of undirected random graphs has centrality [Formula: see text] satisfies a set of recursive distributional equations, which can be analytically characterized and efficiently solved using a population dynamics algorithm. We test our solution on ensembles of Erdős-Rényi and Scale Free networks in the locally tree-like regime, with excellent agreement. The analytical distribution of centrality for the configuration model conditioned on the degree of each node can be employed as a benchmark to identify nodes of empirical networks with over- and underexpressed centrality relative to a null baseline. We also provide an approximate formula based on a rank-[Formula: see text] projection that works well if the network is not too sparse, and we argue that an extension of our method could be efficiently extended to tackle analytical distributions of other centrality measures such as PageRank for directed networks in a transparent and user-friendly way.

Natural Gradient Variational Bayes Without Fisher Matrix Analytic Calculation and Its Inversion

This article introduces a method for efficiently approximating the inverse of the Fisher information matrix, a crucial step in achieving effective variational Bayes inference. A notable aspect of our approach is the avoidance of analytically computing the Fisher information matrix and its explicit inversion. Instead, we introduce an iterative procedure for generating a sequence of matrices that converge to the inverse of Fisher information. The natural gradient variational Bayes algorithm without analytic expression of the Fisher matrix and its inversion is provably convergent and achieves a convergence rate of order O ( log s / s ) , with s the number of iterations. We also obtain a central limit theorem for the iterates. Implementation of our method does not require storage of large matrices, and achieves a linear complexity in the number of variational parameters. Our algorithm exhibits versatility, making it applicable across a diverse array of variational Bayes domains, including Gaussian approximation and normalizing flow Variational Bayes. We offer a range of numerical examples to demonstrate the efficiency and reliability of the proposed variational Bayes method. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.

Rational-quadratic kernel-based maximum correntropy Kalman filter for the non-Gaussian noises

In this paper, a rational-quadratic kernel-based maximum correntropy Kalman filter (RKMCKF) algorithm is proposed to improve the estimation accuracy for non-Gaussian noise interference. Firstly, the RKMCKF algorithm is derived to eliminate the singular matrix produced by multi-dimensional non-Gaussian noise disturbance. Secondly, the upper limit is analyzed to provide a theoretical range for kernel bandwidth, which is beneficial for the selection of proper kernel bandwidths and boosting the precision of state estimation. Furthermore, the boundness of the state estimation error is verified to manifest the RKMCKF algorithm stability. Finally, under different types of non-Gaussian noise, the proposed RKMCKF algorithm is demonstrated to promote the accuracy of state estimation compared with the conventional Kalman filter, Gaussian sum filter, Huber filter, and maximum correntropy Kalman filter through the simulations.

On the Fourier Coefficients of Powers of a Finite Blaschke Product

Abstract Given a finite Blaschke product $B$ we prove asymptotically sharp estimates on the $\ell ^{\infty }$-norm of the sequence of the Fourier coefficients of $B^{n}$ as $n$ tends to $\infty $. This norm decays as $n^{-1/N}$ for some $N\ge 3$. Furthermore, for every $N\ge 3$, we produce explicitly a finite Blaschke product $B$ with decay $n^{-1/N}$. As an application we construct a sequence of $n\times n$ invertible matrices $T$ with arbitrary spectrum in the unit disk and such that the quantity $|\det{T}|\cdot \|T^{-1}\|\cdot \|T\|^{1-n}$ grows as a power of $n$. This is motivated by Schäffer’s question on norms of inverses.

510 Utilizing python programming for modern genetic evaluation systems

Abstract Advancements in agricultural data collection and methodologies for data analysis offer opportunities for beef cattle producers to improve herd genetics. Genetic evaluation systems (GES) can benefit from modern programming approaches to keep up with these advancements. Python, a fast-developing programming language with scientific computing libraries such as NumPy and SciPy, offers flexibility that allows for quick adoption of superior algorithms and new data sources, as well as easy maintenance of the software over the long-term. This study evaluated the use of Python (v3.12.1) for developing a GES capable of executing all the necessary functions expected of modern implementation of GES. A simulated dataset was generated with the R package AlphaSimR to mimic characteristics of the Angus beef cattle breed. The dataset included pedigree information, phenotypic data, and true breeding values (TBVs) for each animal. Phenotypic traits included birth weight (BWT), weaning gain (WG), and post-weaning gain (PWG), with a total of 976,400 animals simulated across 15 generations. The GES software was developed in house, and included components such as data import, pedigree and data preprocessing, calculating inbreeding, constructing the inverse of the numerator relationship matrix, solver initialization, estimation of breeding value, and writing results. The genetic evaluation employed pedigree-based best linear unbiased prediction (BLUP) and a multiple-trait animal model, using a preconditioned conjugate gradient iteration algorithm to solve the mixed model equations. Convergence was assessed using the residual vector norm divided by the right-hand-side norm, with a threshold to declare convergence set to 1x10-5. The software was run on a Linux server with an Intel(R) Xeon(R) Gold 6242 CPU @ 2.80GHz, 96 GB of RAM, and eight cores. Accuracy was measured using Pearson correlation coefficient between TBVs and estimated breeding values (EBVs). To benchmark computing time and solutions, the same dataset and model were input into a commonly used commercial GES, MiXBLUP (v3.0.1). The solver was the most time-consuming component of the GES, with a run time of 27 min. However, the application of multiprocessing reduced the run time by 72% to 8 min. The Pearson correlation coefficient between EBV and TBV was 0.86, 0.79, and 0.73 (P < 0.01) for BWT, WG, and PWG, respectively. In contrast, MiXBLUP completed breeding value prediction in less than 1 min. The Pearson correlation coefficient between Python and MiXBLUP predictions was 1.0 (P < 0.01). While Python is an ideal language for modern GES, further updates are necessary to optimize performance and operations. Future research will focus on refining computational algorithms, exploring parallel processing techniques, and enhancing user interfaces to ensure easy integration with industry practices.

Static analysis using flexibility disassembly perturbation for material nonlinear problem with uncertainty

Nonlinear finite element analysis of large-scale structures usually requires a lot of calculation cost, because it is necessary to repeatedly inverse the modified stiffness matrix caused by nonlinearity in the calculation process. When considering the uncertainty in materials, the calculation of the nonlinear analysis will be more unbearable. To improve the computational efficiency, this work develops a new method for nonlinear analysis with material uncertainty based on flexibility disassembly perturbation (FDP) approach. The FDP is an algorithm that can quickly calculate the inverse of a stiffness matrix. The basic idea of the proposed method is to introduce the FDP formula into Newton-Raphson iteration method to accelerate the nonlinear iterative calculation. Three numerical examples, one statically determinate structure and two statically indeterminate structures, are used to verify the accuracy and efficiency of the proposed method. The results show that the calculation time of the proposed method is far less than that of the existing complete analysis and combined approximation algorithms. In terms of computational accuracy, for statically determinate structures, the proposed algorithm can obtain exact solutions that are identical to the complete analysis results, while for statically indeterminate structures, the proposed algorithm can obtain approximate solutions that are very close to the complete analysis results.