Machine learning the real discriminant locus
Machine learning the real discriminant locus
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DOI：
10.1016/j.jsc.2022.08.001
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发表时间：
2020-06
期刊：
影响因子：
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通讯作者：
Edgar A. Bernal;J. Hauenstein;D. Mehta;Margaret H. Regan;Tingting Tang
中科院分区：
文献类型：
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作者：
Edgar A. Bernal;J. Hauenstein;D. Mehta;Margaret H. Regan;Tingting Tang
多项式方程的参数化系统出现在科学和工程中的许多应用中，其中真实的解描述例如动力系统的平衡、满足设计约束的联动装置以及计算机视觉中的场景重建。由于不同的参数值可以具有不同数量的真实的解，因此将参数空间分解成其边界形成真实的判别轨迹的区域。本文将定位真实的判别轨迹视为机器学习中的监督分类问题，其目标是确定参数空间上的分类边界，类别是真实的解决方案的数量。本文提出了一种新的采样方法，仔细采样的多维参数空间。在每一个采样点处，同伦延拓被用来获得相应多项式系统的真实的解的数目。机器学习技术，包括最近邻，支持向量分类器，和神经网络被用来有效地近似的真实的判别轨迹。已经学习了真实的判别轨迹的一个应用是开发一种真实的同伦方法，该方法仅跟踪真实的解路径，而不像传统方法那样跟踪所有复杂的解路径。算例表明，该方法可以有效地逼近N= 4仓本模型的平衡点等复杂解的边界，而传统方法难以处理这些边界.
Parameterized systems of polynomial equations arise in many applications in science and engineering with the real solutions describing, for example, equilibria of a dynamical system, linkages satisfying design constraints, and scene reconstruction in computer vision. Since different parameter values can have a different number of real solutions, the parameter space is decomposed into regions whose boundary forms the real discriminant locus. This article views locating the real discriminant locus as a supervised classification problem in machine learning where the goal is to determine classification boundaries over the parameter space, with the classes being the number of real solutions. This article presents a novel sampling method which carefully samples a multidimensional parameter space. At each sample point, homotopy continuation is used to obtain the number of real solutions to the corresponding polynomial system. Machine learning techniques including nearest neighbors, support vector classifiers, and neural networks are used to efficiently approximate the real discriminant locus. One application of having learned the real discriminant locus is to develop a real homotopy method that only tracks real solution paths unlike traditional methods which track all complex solution paths. Examples show that the proposed approach can efficiently approximate complicated solution boundaries such as those arising from the equilibria of the N= 4 Kuramoto model which was previously intractable using traditional methods.