Linear Algebra Research Articles

Quadratic twist of an elliptic curve in a generalized Weierstrass equation over a function field

This paper mainly focuses on the construction of a quadratic twist for an elliptic curve represented in a generalized Weierstrass equation over the field Fq(t). The specific form of the quadratic twist, presented in the generalized Weierstrass equation, is determined by linear algebra approach and discussed in detail.

A Provably Accurate Randomized Sampling Algorithm for Logistic Regression

In statistics and machine learning, logistic regression is a widely-used supervised learning technique primarily employed for binary classification tasks. When the number of observations greatly exceeds the number of predictor variables, we present a simple, randomized sampling-based algorithm for logistic regression problem that guarantees high-quality approximations to both the estimated probabilities and the overall discrepancy of the model. Our analysis builds upon two simple structural conditions that boil down to randomized matrix multiplication, a fundamental and well-understood primitive of randomized numerical linear algebra. We analyze the properties of estimated probabilities of logistic regression when leverage scores are used to sample observations, and prove that accurate approximations can be achieved with a sample whose size is much smaller than the total number of observations. To further validate our theoretical findings, we conduct comprehensive empirical evaluations. Overall, our work sheds light on the potential of using randomized sampling approaches to efficiently approximate the estimated probabilities in logistic regression, offering a practical and computationally efficient solution for large-scale datasets.

Log-concave poset inequalities

We study combinatorial inequalities for various classes of set systems: matroids, polymatroids, poset antimatroids, and interval greedoids. We prove log-concave inequal- ities for counting certain weighted feasible words, which generalize and extend several previous results establishing Mason conjectures for the numbers of independent sets of matroids. Notably, we prove matching equality conditions for both earlier inequalities and our extensions. In contrast with much of the previous work, our proofs are combinatorial and employ nothing but linear algebra. We use the language formulation of greedoids which allows a linear algebraic setup, which in turn can be analyzed recursively. The underlying non- commutative nature of matrices associated with greedoids allows us to proceed beyond polymatroids and prove the equality conditions. As further application of our tools, we rederive both Stanley’s inequality on the number of certain linear extensions, and its equality conditions, which we then also extend to the weighted case.

Linear algebra and congruences for MacMahon’s k-rowed plane partitions

In this paper, we provide an algorithm to detect linear congruences of [Formula: see text], the number of MacMahon’s [Formula: see text]-rowed plane partitions, and give a quantitative result on the nonexistence of Ramanujan-type congruences of the [Formula: see text]-rowed plane partition functions. We also show [Formula: see text] that the number of partitions at most [Formula: see text] parts always admits linear congruences.

A High-Speed Floating Point Matrix Multiplier Implemented in Reconfigurable Architecture

Matrix multiplication is a fundamental operation in computational applications across various domains. This paper introduces a novel reconfigurable co-processor that enhances the efficiency of matrix multiplication by concurrently executing addition and multiplication operations upon matrix elements of different sizes. The proposed design aims to reduce computation time and improve efficiency for matrix multiplication equations. Experimental evaluations were conducted on matrices of different sizes to demonstrate the effectiveness of the processor. The results reveal substantial improvements in both time and efficiency when compared to traditional approaches. The reconfigurable transformation processor harnesses parallel processing capabilities, enabling the simultaneous execution of addition and multiplication operations by partitioning input matrices into smaller submatrices and performing parallel computations, thus the processor achieves faster results. Additionally, the design incorporates configurable arithmetic units that dynamically adapt to matrix characteristics, further optimizing performance. The experimental evaluations provide evidence of reduction in computation time and improvement in efficiency. present significant benefits over traditional sequential methods. This makes this co-processor ideally fit for domains that require intensive linear algebra computations such as computer vision, machine learning, and signal processing.

Closure Operations on Intuitionistic Linear Algebras

Abstract In this paper, we introduce the notions of radical filters and extended filters of Intuitionistic Linear algebras (IL-algebras for short) and give some of their properties. The notion of closure operation on an IL-algebra is also introduced as well as the study of some of their main properties. The radical of filters and extended filters are examples of closure operations among several others provided. The class of stable closure operations on an IL-algebra L is used to study the unifying properties of some subclasses of the lattice of filters of L. In particular, we obtain that for a stable closure operation c on an IL-algebra, the collection of c-closed elements of its lattice of filters forms a complete Heyting algebra. Hyperarchimedean IL-algebras are also characterized using closure operations. It is shown that the image of a semi-prime closure operation on an IL-algebra is isomorphic to a quotient IL-algebra. Some properties of the quotients induced by closure operations on an IL-algebra are explored.

Monte Carlo on manifolds in high dimensions

We introduce an efficient numerical implementation of a Markov Chain Monte Carlo method to sample a probability distribution on a manifold (introduced theoretically in Zappa, Holmes-Cerfon, Goodman (2018) [53]), where the manifold is defined by the level set of constraint functions, and the probability distribution may involve the pseudodeterminant of the Jacobian of the constraints, as arises in physical sampling problems. The algorithm is easy to implement and scales well to problems with thousands of dimensions and with complex sets of constraints provided their Jacobian retains sparsity. The algorithm uses direct linear algebra and requires a single matrix factorization per proposal point, which enhances its efficiency over previously proposed methods but becomes the computational bottleneck of the algorithm in high dimensions. We test the algorithm on several examples inspired by soft-matter physics and materials science to study its complexity and properties.

QuTree: A tree tensor network package.

We present QuTree, a C++ library for tree tensor network approaches. QuTree provides class structures for tensors, tensor trees, and related linear algebra functions that facilitate the fast development of tree tensor network approaches such as the multilayer multiconfigurational time-dependent Hartree approach or the density matrix renormalization group approach and its various extensions. We investigate the efficiency of relevant tensor and tensor network operations and show that the overhead for managing the network structure is negligible, even in cases with a million leaves and small tensors. QuTree focuses on providing simple, high-level routines while retaining easy access to the backend to facilitate novel developments. We demonstrate the capabilities of the package by computing the eigenstates of coupled harmonic oscillator Hamiltonians and performing random circuit simulations on a virtual quantum computer.

Mathematics in Data Science and Artificial Intelligence

Mathematics is a discipline that focuses on structure, order, and relation, derived from counting, measuring, and characterizing object shapes. Mathematics is necessary for professions in data science since machine learning algorithms, conducting analyses, and drawing conclusions from data all require it. A key component of data science is math. It can support problem-solving, model performance optimization, and the interpretation of complex data to address business-related queries. The technology known as artificial intelligence (AI) has come to revolutionize many facets of our existence. Mathematics plays a fundamental part in the astounding advances and capabilities of artificial intelligence. Mathematics contains various branches like algebra, geometry, Trigonometry, Calculus, Statistics and Probability. The foundation of mathematics gives artificial intelligence (AI) systems the ability to reason, learn, and make wise judgments. This article examines the relevance and use of mathematics in artificial intelligence. Large-scale data processing, analysis, and interpretation are made possible by machines thanks to mathematics, which forms the foundation of AI models and algorithms. Developing machine learning algorithms requires an understanding of concepts from statistics, probability theory, calculus, and linear algebra. These algorithms recognize patterns, forecast outcomes, and categorize data using mathematical equations and functions.

Role of Mathematics in Data Science - Machine learning

The mathematical concepts help with identifying patterns and algorithmic development. Mathematics is an important part of many modern scientific areas. It is also a field of major significance in data science. Mainly because of the concepts they propose and the tools they allow, statistics and arithmetic have such an essential role in data science. A specific type of mathematical reasoning is needed for analysing data, in addition to the basic concepts of calculus, discrete mathematics, and linear algebra. To understand how to apply such algorithms in data science, one must have a solid understanding of many different statistical and probability theory ideas. Modern data science methods have a strong mathematical basis for almost all of their techniques, including machine learning.

Algorithm 1039: Automatic Generators for a Family of Matrix Multiplication Routines with Apache TVM

We explore the utilization of the Apache TVM open source framework to automatically generate a family of algorithms that follow the approach taken by popular linear algebra libraries, such as GotoBLAS2, BLIS, and OpenBLAS, to obtain high-performance blocked formulations of the general matrix multiplication ( gemm ). In addition, we fully automatize the generation process by also leveraging the Apache TVM framework to derive a complete variety of the processor-specific micro-kernels for gemm . This is in contrast with the convention in high-performance libraries, which hand-encode a single micro-kernel per architecture using Assembly code. In global, the combination of our TVM-generated blocked algorithms and micro-kernels for gemm (1) improves portability, maintainability, and, globally, streamlines the software life cycle; (2) provides high flexibility to easily tailor and optimize the solution to different data types, processor architectures, and matrix operand shapes, yielding performance on a par (or even superior for specific matrix shapes) with that of hand-tuned libraries; and (3) features a small memory footprint.

Hermitian Dynamic Mode Decomposition - Numerical Analysis and Software Solution

The Dynamic Mode Decomposition (DMD) is a versatile and increasingly popular method for data driven analysis of dynamical systems that arise in a variety of applications in, e.g., computational fluid dynamics, robotics or machine learning. In the framework of numerical linear algebra, it is a data driven Rayleigh-Ritz procedure applied to a DMD matrix that is derived from the supplied data. In some applications, the physics of the underlying problem implies hermiticity of the DMD matrix, so the general DMD procedure is not computationally optimal. Furthermore, it does not guarantee important structural properties of the Hermitian eigenvalue problem and may return non-physical solutions. This paper proposes a software solution to the Hermitian (including the real symmetric) DMD matrices, accompanied with a numerical analysis that contains several fine and instructive numerical details. The eigenpairs are computed together with their residuals, and perturbation theory provides error bounds for the eigenvalues and eigenvectors. The software is developed and tested using the LAPACK package.

A LAPACK Implementation of the Dynamic Mode Decomposition

The Dynamic Mode Decomposition (DMD) is a method for computational analysis of nonlinear dynamical systems in data driven scenarios. Based on high fidelity numerical simulations or experimental data, the DMD can be used to reveal the latent structures in the dynamics or as a forecasting or a model order reduction tool. The theoretical underpinning of the DMD is the Koopman operator on a Hilbert space of observables of the dynamics under study. This paper describes a numerically robust and versatile variant of the DMD and its implementation using the state-of-the-art dense numerical linear algebra software package LAPACK . The features of the proposed software solution include residual bounds for the computed eigenpairs of the DMD matrix, eigenvectors refinements and computation of the eigenvectors of the Exact DMD, compressed DMD for efficient analysis of high dimensional problems that can be easily adapted for fast updates in a streaming DMD. Numerical analysis is the bedrock of numerical robustness and reliability of the software, that is tested following the highest standards and practices of LAPACK . Important numerical topics are discussed in detail and illustrated using numerous numerical examples.

Teaching Matrix Multiplication with Impactful and Productive Meaning

What is the meaning of matrix multiplication? In this article, I outline a sequence of lessons for an introductory linear algebra course to teach matrix multiplication with impactful and productive meaning. These lessons are part of a broader linear algebra curriculum I have developed centered around students developing core mathematical mental actions such as meaning-making, symbolizing, interpreting, and justifying. For each lesson, I provide goals, a lesson plan, and an explanation of my instructional choices.

Inertia of partial transpose of positive semidefinite matrices

We show that the partial transpose of 9×9 positive semidefinite matrices do not have inertia (4,1,4) and (3,2,4) . It solves an open problem in ‘LINEAR AND MULTILINEAR ALGEBRA. Changchun Feng et al, 2022’. We apply our results to construct some inertia, as well as present the list of all possible inertia of partial transpose of 12×12 positive semidefinite matrices.

Accurate Computations with Block Checkerboard Pattern Matrices

In this work, block checkerboard sign pattern matrices are introduced and analyzed. They satisfy the generalized Perron–Frobenius theorem. We study the case related to total positive matrices in order to guarantee bidiagonal decompositions and some linear algebra computations with high relative accuracy. A result on intervals of checkerboard matrices is included. Some numerical examples illustrate the theoretical results.

The Role of Higher Mathematics in Public Security Political Work Specialty Based on the New Liberal Arts

The construction of new liberal arts is a systematic education reform project, which is of great significance to improve the quality of college liberal arts talents training in the new era. Under the background of new liberal arts construction, higher mathematics plays an indispensable role in the political work of public security. In fact, statistics, linear algebra and probability theory are widely used and play an important role in the public security cause. In addition, based on the follow-up study of advanced mathematics in liberal arts, students who graduate from the police offi cer will make a great role in crime analysis and prediction case analysis and case analysis of public safety management. Finally, based on the concept of OBE, the paper puts forward the refl ection of designing the teaching content of higher mathematics in public security political work specialty.

Practice on Construction of Linear Algebra Question Bank Based on ChaoXing Fanya Platform

In recent years,in all kinds of colleges and universities in our country,the ChaoXing Fanya platform has been widely used in informationize teaching of the courses.Relying on the ChaoXing Fanya platform,closely focusing on the linear algebra curriculum standards,the teaching team established a linear algebra question bank,adopted online and offl ine mixed teaching,combined paperless testing with process evaluation,which better refl ects the goal of examination separated from teaching.

Avoiding breakdown in incomplete factorizations in low precision arithmetic

The emergence of low precision floating-point arithmetic in computer hardware has led to a resurgence of interest in the use of mixed precision numerical linear algebra. For linear systems of equations, there has been renewed enthusiasm for mixed precision variants of iterative refinement. We consider the iterative solution of large sparse systems using incomplete factorization preconditioners. The focus is on the robust computation of such preconditioners in half precision arithmetic and employing them to solve symmetric positive definite systems to higher precision accuracy; however, the proposed ideas can be applied more generally. Even for well-conditioned problems, incomplete factorizations can break down when small entries occur on the diagonal during the factorization. When using half precision arithmetic, overflows are an additional possible source of breakdown. We examine how breakdowns can be avoided and we implement our strategies within new half precision Fortran sparse incomplete Cholesky factorization software. Results are reported for a range of problems from practical applications. These demonstrate that, even for highly ill-conditioned problems, half precision preconditioners can potentially replace double precision preconditioners, although unsurprisingly this may be at the cost of additional iterations of a Krylov solver.

Quantum Communication Complexity of Linear Regression

Quantum computers may achieve speedups over their classical counterparts for solving linear algebra problems. However, in some cases—such as for low-rank matrices—dequantized algorithms demonstrate that there cannot be an exponential quantum speedup. In this work, we show that quantum computers have provable polynomial and exponential speedups in terms of communication complexity for some fundamental linear algebra problems if there is no restriction on the rank. We mainly focus on solving linear regression and Hamiltonian simulation. In the quantum case, the task is to prepare the quantum state of the result. To allow for a fair comparison, in the classical case, the task is to sample from the result. We investigate these two problems in two-party and multiparty models, propose near-optimal quantum protocols, and prove quantum/classical lower bounds. In this process, we propose an efficient quantum protocol for quantum singular value transformation, which is a powerful technique for designing quantum algorithms. We feel this will be helpful in developing efficient quantum protocols for many other problems.