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The Journal of Machine Learning Research 2022年第1期23卷 15156-15203页

作者： Agniva Chowdhury Gregory Dexter Palma London Haim Avron Petros Drineas Computer Science and Mathematics Division Oak Ridge National Laboratory Oak Ridge TN Department of Computer Science Purdue University West Lafayette IN Operations Research and Information Engineering Cornell University Ithaca NY School of Mathematical Sciences Tel Aviv University Tel Aviv Israel

Linear programming (LP) is an extremely useful tool which has been successfully applied to solve various problems in a wide range of areas, including operations research, engineering, economics, or even more abstract mathematical areas such as combinatorics. It is also used in many machine learning applications, such as ℓ1-regularized SVMs, basis pursuit, nonnegative matrix factorization, etc. Interior Point Methods (IPMs) are one of the most popular methods to solve LPs both in theory and in practice. Their underlying complexity is dominated by the cost of solving a system of linear equations at each iteration. In this paper, we consider both feasible and infeasible IPMs for the special case where the number of variables is much larger than the number of constraints. Using tools from Randomized Linear algebra, we present a preconditioning technique that, when combined with the iterative solvers such as Conjugate Gradient or Chebyshev Iteration, provably guarantees that IPM algorithms (suitably modified to account for the error incurred by the approximate solver), converge to a feasible, approximately optimal solution, without increasing their iteration complexity. Our empirical evaluations verify our theoretical results on both real-world and synthetic data.

关键词： randomized linear algebra linear programming interior point methods

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The cryptologic characteristics of circulant matrices

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International Journal of Innovative Computing and applications 2021年第5-6期12卷 248-254页

作者： Han, Haiqing Zhu, Siru Li, Qin He, Yanping Wang, Xiao Wang, Yuxia School of Computer Wuhan University Hubei Wuhan430072 China School of Mathematics and Physics Hubei Polytechnic University Hubei Huangshi435003 China Air Force Early Warning Academy of PLA Hubei Province Wuhan City430072 China Normal Department Hubei Polytechnic University Hubei Province Huangshi435003 China

A 4 × 4 invertible circulant matrix on GF(28) can represent the Mixcolumn operation of AES, which plays an important role as a confusion operation. Starting from the analysis of the Mixcolumns operation of AES, we have mainly research the properties of circulant matrix over finite field, and present a novel algorithm that generates the 4 × 4 circulant inverse matrices and the 4 × 4 circulant matrices with the maximal branch number in this paper. At last, some characteristics of the orthormorphic or symmetrical circulant matrices to arrive at the maximal branch number have been discussed in this paper. An algorithm for generating the 4 × 4 orthormorphic circulant matrix with the maximal branch number is also obtained on the finite field. Furthermore, the conclusion is gained that the symmetrical matrix with maximal branch number does not exist. Copyright © 2021 Inderscience Enterprises Ltd.

关键词： Matrix algebra

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On Newton polygons techniques and factorization of polynomials over Henselian fields

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JOURNAL OF algebra AND ITS applications 2020年第10期19卷

作者： El Fadil, Lhoussain Sidi Mohamed ben Abdellah Univ Fac Sci Dhar El Mahraz Fes Morocco

Let (K, nu) be a valued field, where nu is a rank-one discrete valuation, with valuation ring R. The goal of this paper is to investigate some basic concepts of Newton polygon techniques of a monic polynomial f (x) is an element of R[x];namely, theorem of the product, of the polygon, and of the residual polynomial, in such way that improves that given in [D. Cohen, A. Movahhedi and A. Salinier, Factorization over local fields and the irreducibility of generalized difference polynomials, Mathematika 47 (2000) 173-196] and generalizes that given in [J. Guardia, J. Montes and E. Nart, Newton polygons of higher order in algebraic number theory, Trans. Amer. Math. Soc. 384(1) (2012) 361-416] to any rank-one valued field.

关键词： Valued fields Henselization Newton polygon theorem of the polygon theorem of the residual polynomial prime ideal factorization

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Sparse resultant based minimal solvers in computer vision and their connection with the action matrix

arXiv

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arXiv 2023年

作者： Bhayani, Snehal Heikkilä, Janne Kukelova, Zuzana Center for Machine Vision and Signal Analysis Faculty of Information Technology and Electrical Engineering Oulu Finland Visual Recognition Group Department of Cybernetics Czech Technical University Prague Czech Republic

Many computer vision applications require robust and efficient estimation of camera geometry from a minimal number of input data measurements, i.e., solving minimal problems in a RANSAC framework. Minimal problems are usually formulated as complex systems of sparse polynomial equations. The systems usually are overdetermined and consist of polynomials with algebraically constrained coefficients. Most state-of-the-art efficient polynomial solvers are based on the action matrix method that has been automated and highly optimized in recent years. On the other hand, the alternative theory of sparse resultants and Newton polytopes has been less successful for generating efficient solvers, primarily because the polytopes do not respect the constraints on the coefficients. To tackle this challenge, in this paper, we propose a simple iterative scheme to test various subsets of the Newton polytopes and search for the most efficient solver. Moreover, we propose to use an extra polynomial with a special form to further improve the solver efficiency via a Schur complement computation. We show that for some camera geometry problems our extra polynomial-based method leads to smaller and more stable solvers than the state-of-the-art Gröbner basis-based solvers. The proposed method can be fully automated and incorporated into existing tools for automatic generation of efficient polynomial solvers. It provides a competitive alternative to popular Gröbner basis-based methods for minimal problems in computer vision. Additionally, we study the conditions under which the minimal solvers generated by the state-of-the-art action matrix-based methods and the proposed extra polynomial resultant-based method, are equivalent. Specifically we consider a step-by-step comparison between the approaches based on the action matrix and the sparse resultant, followed by a set of substitutions, which would lead to equivalent minimal solvers. © 2023, CC BY.

关键词： Matrix algebra

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On A Novel Security Scheme for The Encryption and Decryption Of 2×2 Fuzzy Matrices with Rational Entries Based on The algebra of Neutrosophic Integers and El-Gamal Crypto-System

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Neutrosophic Sets and Systems 2023年 54卷 19-32页

作者： Abobala, Mohammad Allouf, Ali Tishreen University Department Of Mathematics Latakia Syria Tishreen University Faculty Of computer engineering and automatio Latakia Syria

The main goal behind mathematical cryptography is to keep messages and multimedia messages secret at a time when modern means of communication have spread and become very diverse. Fuzzy matrices as strong tools which was defined to deal with incomplete and uncertain data and many relationships in real life problems especially those which are related to images and graphs, may considered as important subjects for secret information and communication. The aim of this research paper is to present a new model and method for encrypting 2×2 fuzzy matrices using the basic concepts in neutrosophic number theory and El Gamal algorithm in cryptography, where we generalize El Gamal algorithm to become applicable to the ring of neutrosophic integer numbers that represents the studied fuzzy matrices. On the other hand, we study the applications of the novel algorithm to the encryption and decryption of some fuzzy relations represented in terms of fuzzy *** addition, we illustrate many examples to clarify the validity of the new algorithm. © 2023,Neutrosophic Sets and Systems. All Rights Reserved.

关键词： Cryptography

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Solving Boolean polynomial systems by parallelizing characteristic set method for cyber-physical systems

Solving Boolean polynomial systems by parallelizing characte...

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作者： Zhao, Juan Zhu, Min Li, Xiaoyong Huang, Zhenyu Li, Jincai Song, Junqiang College of Meteorology and Oceanography National University of Defense Technology Changsha China College of Computer Science and Technology National University of Defense Technology Changsha China SKLOIS Institute of Information Engineering Chinese Academy of Sciences Beijing China

Many cyber-attach schemes and coding models established by algebra tools are build to address the problem of security of cyber-pysical systems (CPS). As an important field of algebra computing, Boolean Polynomial System Solving (PoSSo) problem plays a very important role in many algebra applications. In this article, we propose an efficient Parallel Boolean Characteristic Set method (PBCS) under the high-performance computing environment to improve the efficiency of solving Boolean polynomial systems. The PBCS is implemented based on the state-of-the-art Boolean Characteristic Set method (BCS). It adopts a master-slave parallel pattern, and distributes tasks based on the polynomial sets after initial zero decomposition. We design a strategy of dynamically reallocating tasks to ameliorate load imbalance, which is caused by dynamical zero decomposition of polynomials. Furthermore, we improve its performance by optimizing the parameter settings of PBCS, including the maximum number of polynomial branches that trigger the dynamic allocation policy and the scheduling time. Experimental results with solving several Boolean polynomial systems confirm that PBCS is efficient and scalable, especially for the equations generating from stream ciphers that have block triangular structure. Moreover, the method also has good scalability. It shows a stable speedup as well even extending to the size of thousands of CPU cores. © 2020 John Wiley & Sons Ltd.

关键词： Cyber Physical System

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New Type of Gegenbauer-Jacobi-Hermite Monogenic polynomials and Associated Continuous Clifford Wavelet Transform Some Monogenic Clifford polynomials and Associated Wavelets

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ACTA APPLICANDAE MATHEMATICAE 2020年第1期170卷 1-35页

作者： Arfaoui, Sabrine Ben Mabrouk, Anouar Cattani, Carlo Univ Monastir Dept Math Algebra Number Theory & Nonlinear Anal Lab LR18ES Fac Sci Monastir 5000 Tunisia Univ Tabuk Dept Math Fac Sci Tabuk Saudi Arabia Univ Kairouan Higher Inst Appl Math & Comp Sci Dept Math Kairouan 3100 Tunisia Tuscia Univ Engn Sch DEIM Tuscia Italy

Recently 3D image processing has interested researchers in both theoretical and applied fields and thus has constituted a challenging subject. Theoretically, this needs suitable functional bases that are easy to implement by the next. It holds that Clifford wavelets are main tools to achieve this necessity. In the present paper we intend to develop some new classes of Clifford wavelet functions. Some classes of new monogenic polynomials are developed firstly from monogenic extensions of 2-parameters Clifford weights. Such classes englobe the well known Jacobi, Gegenbauer and Hermite ones. The constructed polynomials are next applied to develop new Clifford wavelets. Reconstruction and Fourier-Plancherel formulae have been proved. Finally, computational examples are developed provided with graphical illustrations of the Clifford mother wavelets in some cases. Some graphical illustrations of the constructed wavelets have been provided and finally concrete applications in biofields have been developed dealing with EEG/ECG and Brain image processing.

关键词： Continuous wavelet transform Clifford analysis Clifford Fourier transform Fourier-Plancherel monogenic polynomials EEG ECG Brain images

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A RIEMANN-HILBERT APPROACH TO THE PERTURBATION theory FOR ORTHOGONAL polynomials: applications TO NUMERICAL LINEAR algebra AND RANDOM MATRIX theory

arXiv

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arXiv 2021年

作者： Ding, Xiucai Trogdon, Thomas University of California Davis United States University of Washington SeattleWA United States

We establish a new perturbation theory for orthogonal polynomials using a Riemann–Hilbert approach and consider applications in numerical linear algebra and random matrix theory. This new approach shows that the orthogonal polynomials with respect to two measures can be effectively compared using the difference of their Stieltjes transforms on a suitably chosen contour. Moreover, when two measures are close and satisfy some regularity conditions, we use the theta functions of a hyperelliptic Riemann surface to derive explicit and accurate expansion formulae for the perturbed orthogonal polynomials. In contrast to other approaches, a key strength of the methodology is that estimates can remain valid as the degree of the polynomial grows. The results are applied to analyze several numerical algorithms from linear algebra, including the Lanczos tridiagonalization procedure, the Cholesky factorization and the conjugate gradient algorithm. As a case study, we investigate these algorithms applied to a general spiked sample covariance matrix model by considering the eigenvector empirical spectral distribution and its limits. For the first time, we give precise estimates on the output of the algorithms, applied to this wide class of random matrices, as the number of iterations diverges. In this setting, beyond the first order expansion, we also derive a new mesoscopic central limit theorem for the associated orthogonal polynomials and other quantities relevant to numerical *** Codes 42C05, 65F10, 60B20 Copyright © 2021, The Authors. All rights reserved.

关键词： polynomials

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THE RELATIVE MANIN–MUMFORD CONJECTURE

arXiv

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arXiv 2023年

作者： Gao, Ziyang Habegger, Philipp Institute of Algebra Number Theory and Discrete Mathematics Leibniz University Hannover Welfengarten 1 Hannover30167 Germany Department of Mathematics and Computer Science University of Basel Spiegelgasse 1 Basel4051 Switzerland

MSC Codes 11G30, 11G50, 14G05, 14G25We prove the Relative Manin–Mumford Conjecture for families of abelian varieties in characteristic 0. We follow the Pila–Zannier method to study special point problems, and we use the Betti map which goes back to work of Masser and Zannier in the case of curves. The key new ingredients compared to previous applications of this approach are a height inequality proved by both authors of the current paper and Dimitrov, and the first-named author’s study of certain degeneracy loci in subvarieties of abelian schemes. We also strengthen this result and prove a criterion for torsion points to be dense in a subvariety of an abelian scheme over C. The Uniform Manin–Mumford Conjecture for curves embedded in their Jacobians was first proved by Kühne. We give a new proof, as a corollary to our main theorem, that does not use equidistribution. © 2023, CC BY.

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On multivariate polynomials with many roots over a finite grid

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JOURNAL OF algebra AND ITS applications 2021年第8期20卷 2150136-2150136页

作者： Geil, Olav Aalborg Univ Dept Math Sci DK-9100 Aalborg Denmark

In this paper, we consider roots of multivariate polynomials over a finite grid. When given information on the leading monomial with respect to a fixed monomial ordering, the footprint bound [Footprints or generalized Bezout's theorem, IEEE Trans. Inform. theory 46(2) (2000) 635-641, On (or in) Dick Blahut's 'footprint', Codes, Curves Signals (1998) 3-9] provides us with an upper bound on the number of roots, and this bound is sharp in that it can always be attained by trivial polynomials being a constant times a product of an appropriate combination of terms consisting of a variable minus a constant. In contrast to the one variable case, there are multivariate polynomials attaining the footprint bound being not of the above form. This even includes irreducible polynomials. The purpose of the paper is to determine a large class of polynomials for which only the mentioned trivial polynomials can attain the bound, implying that to search for other polynomials with the maximal number of roots one must look outside this class.

关键词： Finite field finite grid footprint bound multivariate polynomial root variety

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