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On the acceleration of gradient methods: the triangle steepest descent method

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arXiv 2025年

作者： Shen, Ya Li, Qing-Na Dai, Yu-Hong School of Mathematics and Statistics Beijing Institute of Technology Beijing100081 China School of Mathematics and Statistics Beijing Key Laboratory on MCAACI/Key Laboratory of Mathematical Theory and Computation in Information Security Beijing Institute of Technology Beijing100081 China LSEC ICMSEC AMSS Chinese Academy of Sciences Beijing100190 China School of Mathematical Sciences University of Chinese Academy of Sciences Beijing100049 China

The gradient type of methods has been a competitive choice in solving large scale problems arising from various applications such as machine learning. However, there is still space to accelerate the gradient methods. To this end, in this paper, we pay attention to the cyclic steepest descent method (CSD), and prove that the CSD method has a gradient subsequence that is R-superlinearly convergent for the 2-dimensional strictly convex quadratic case. Moreover, we propose a new gradient method called triangle steepest descent method (TSD) which has a parameter j to control the number of cycles. This method is motivated by utilizing a geometric property of the steepest descent method (SD) method to get around the zigzag behavior. We show that the TSD method is at least R-linearly convergent for strictly convex quadratic problems. The advantage of the TSD method is that it is not sensitive to the condition number of a strictly convex quadratic problem. For example, it performs better than other competitive gradient methods when the condition number reaches 1e20 or 1e100 for some strictly convex quadratic problems. Extensive numerical results verify the efficiency of the TSD method compared to other types of gradient *** Codes 90C20 \\and 90C25 Copyright © 2025, The Authors. All rights reserved.

关键词： Steepest descent method

A NONCOMMUTATIVE APPROACH TO THE SCHUR POSITIVITY OF CHROMATIC SYMMETRIC FUNCTIONS

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arXiv 2023年

作者： Thibon, Jean-Yves Wang, David G.L. LIGM Université Gustave-Eiffel CNRS ENPC ESIEE-Paris 5 Boulevard Descartes Champs-sur-Marne Marne-la-Vallée77454 Cedex 2 France School of Mathematics and Statistics MIIT Key Laboratory of Mathematical Theory and Computation in Information Security Beijing Institute of Technology Beijing102400 China

We obtain the Schur positivity of spider graphs of the forms S(a, 2, 1) and S(a, 4, 1), which are considered to have the simpliest structures for which the Schur positivity was unknown. The proof outline has four steps. First, we find noncommutative analogs for the chromatic symmetric functions of the spider graphs S(a, b, 1). Secondly, we expand the analogs under the Λ- and R-bases, whose commutative images are the elementary and skew Schur symmetric functions, respectively. Thirdly, we recognize the Schur coefficients via the Littlewood-Richardson rule in terms of norms of multisets of Yamanouchi words. At last we establish the Schur positivity combinatorially together with the aid of computer *** Codes 05E05, 05C05, 05A15 Copyright © 2023, The Authors. All rights reserved.

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ON SCATTERING ASYMPTOTICS FOR THE 2D CUBIC RESONANT SYSTEM

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arXiv 2022年

作者： Yang, Kailong Zhao, Zehua Chongqing National Center for Applied Mathematics Chongqing Normal University Chongqing China Department of Mathematics University of Maryland United States Department of Mathematics and Statistics Beijing Institute of Technology MIIT Key Laboratory of Mathematical Theory and Computation in Information Security China

In this paper, we prove scattering asymptotics for the 2D (discrete dimension) cubic resonant system. This scattering result was used in Zhao [39] as an assumption to obtain the scattering for cubic NLS on R2 × T2 in H1 space. Moreover, the 1D analogue is proved in Yang-Zhao [36]. Though the scheme is also tightly based on Dodson [12], the 2D case is more complicated which causes some new difficulties. One obstacle is the failure of ‘l2-estimate’ for the cubic resonances in 2D (we also discuss it in this paper, which may have its own interests). To fix this problem, we establish weaker estimates and exploit the symmetries of the resonant system to modify the proof of [36]. At last, we make a few remarks on the research line of ‘long time dynamics for NLS on waveguides’. Copyright © 2022, The Authors. All rights reserved.

关键词： Waveguides

On Strichartz Estimates for Many-Body Schrödinger Equation in the Periodic Setting

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SSRN

SSRN 2023年

作者： Yu, Xueying Zhao, Zehua Zheng, Jiqiang Department of Mathematics University of Washington C138 Padelford Hall Box 354350 SeattleWA98195 United States Department of Mathematics and Statistics Beijing Institute of Technology Beijing China MIIT Key Laboratory of Mathematical Theory and Computation in Information Security Beijing China Institute of Applied Physics and Computational Mathematics Beijing100088 China

In this paper, we prove Strichartz estimates for N-body Schrödinger equations in the periodic setting, specifically on tori T d , where d ≥ 3. The results hold for both rational and irrational tori, and for small interacting potentials in a certain sense. Our work is based on the standard Strichartz estimate for Schr¨odinger operators on periodic domains, as developed by Bourgain-Demeter [2], and can also be extended to the waveguide case using Barron’s recent result [1]. This result can be regarded as a periodic analogue of Hong [20]. © 2023, The Authors. All rights reserved.

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Graded dimensions and monomial bases for the cyclotomic quiver hecke superalgebras

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arXiv 2021年

作者： Hu, Jun Shi, Lei MIIT Key Laboratory of Mathematical Theory and Computation in Information Security Beijing Institute of Technology Beijing102488 China School of Mathematical and Statistics Beijing Institute of Technology Beijing100081 China

MSC Codes 20C08In this paper we derive a closed formula for the (Z × Z2)-graded dimension of the cyclotomic quiver Hecke superalgebra RΛ(β) associated to an arbitrary Cartan superdatum (A, P, Π, Π∨), polynomials (Qi,j(x1, x2))i,j∈I, β ∈ Q+n and Λ ∈ P +. As applications, we obtain a necessary and sufficient condition for which e(ν) 6= 0 in RΛ(β). We construct an explicit monomial basis for the bi-weight space e(νe)RΛ(β)e(νe), where νe is a certain specific ntuple defined in (1.4). In particular, this gives rise to a monomial basis for the cyclotomic odd nilHecke algebra. Finally, we consider the case when β = α1 +α2 +· · ·+αn with α1, · · ·, αn distinct. We construct an explicit monomial basis of RΛ(β) and show that it is indecomposable in this case. Copyright © 2021, The Authors. All rights reserved.

关键词： Algebra

An Efficient Method for Sample Adversarial Perturbations against Nonlinear Support Vector Machines

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arXiv 2022年

作者： Su, Wen Li, Qingna School of Mathematics and Statistics Beijing Institute of Technology Beijing China School of Mathematics and Statistics Beijing Key Laboratory on MCAACI Key Laboratory of Mathematical Theory and Computation in Information Security Beijing Institute of Technology Beijing China

Adversarial perturbations have drawn great attentions in various machine learning models. In this paper, we investigate the sample adversarial perturbations for nonlinear support vector machines (SVMs). Due to the implicit form of the nonlinear functions mapping data to the feature space, it is difficult to obtain the explicit form of the adversarial perturbations. By exploring the special property of nonlinear SVMs, we transform the optimization problem of attacking nonlinear SVMs into a nonlinear KKT system. Such a system can be solved by various numerical methods. Numerical results show that our method is efficient in computing adversarial perturbations. Copyright © 2022, The Authors. All rights reserved.

关键词： Numerical methods

Observer Design for a Reaction-Diffusion Equation with Delayed Boundary Measurement

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Observer Design for a Reaction-Diffusion Equation with Delay...

International Conference on Control and Robots (ICCR)

作者： Jing Zhang Wen Kang School of Automation and Electrical Engineering University of Science and Technology Beijing Beijing China Shunde Innovation School University of Science and Technology Beijing Foshan China MIIT Key Laboratory of Mathematical Theory and Computation in Information Security School of Mathematics and Statistics Beijing Institute of Technology Beijing China

This paper addresses the problem of observer design for a reaction-diffusion equation with delayed boundary measurement. The predictor method is employed for the delay compensation. The reaction coefficient of the plant is considered, which may lead to the instability. To obtain the stability result, the backstepping transformation is introduced to determine the observer gains. Linear matrix inequalities (LMIs) conditions are derived to guarantee the exponential stability of the error system. The numerical simulation is provided to validate the proposed results.

关键词： Backstepping Observers Numerical simulation mathematical models Stability analysis Delays Linear matrix inequalities

ON THE DECAY PROPERTY OF THE CUBIC FOURTH-ORDER SCHRÖDINGER EQUATION

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arXiv 2022年

作者： Yu, Xueying Yue, Haitian Zhao, Zehua Department of Mathematics University of Washington C138 Padelford Hall Box 354350 SeattleWA98195 United States Institute of Mathematical Sciences ShanghaiTech University Pudong Shanghai China Department of Mathematics and Statistics Beijing Institute of Technology MIIT Key Laboratory of Mathematical Theory and Computation in Information Security Beijing China

In this short paper, we prove that the solution of the cubic fourth-order Schrödinger equation (4NLS) on d (5 ≤ d ≤ 8) enjoys the same (pointwise) decay property as its linear solution does. This result is proved via a bootstrap argument based on the corresponding global result Pausader [24]. This result can be extended to more general dispersive equations (including some more 4NLS models) with scattering asymptotics. Copyright © 2022, The Authors. All rights reserved.

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On some families of binary codes

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information Processing Letters 2026年 191卷

作者： Zihui Liu Key Laboratory of Mathematical Theory and Computation in Information Security Department of Mathematics Beijing Institute of Technology Beijing 100081 China

We give further results on the weight distributions of the two families of binary codes recently constructed by simplicial complexes by (Wu, Lee, 2020), and show that the converse of the above results is also correct, that is, the binary codes with such weight distributions properties must be these two families of codes. Based on the above results, we also construct another family of binary self-orthogonal codes and present their separating properties and applications to the secret sharing scheme, cryptography and other aspects of information security.

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