几类约束张量逼近的理论与数值算法研究
发布时间：2018-07-25 14:31
【摘要】：约束张量逼近问题是数值代数领域研究和探讨的重要课题.它在盲源分离、高阶统计、机器学习和谐波恢复等领域有着广泛的应用.本文系统地研究了几类约束张量逼近问题的理论和数值算法.第二章,研究了对称张量多重线性低秩逼近问题(?)我们首先将该问题重构成黎曼流形上的极大化问题,再将欧式空间上的谱共轭梯度法推广至黎曼流形上,设计了黎曼流形上的谱共轭梯度法(RSCG),接着利用RSCG方法求解等价的极大化问题,最后用数值例子验证了新方法的可行性和有效性。第三章,研究了Hankel张量逼近问题(?)利用半正定Hankel矩阵的范德蒙分解将强Hankel张量逼近问题转化为无约束优化问题,并借助非线性共轭梯度法进行求解.我们设计Dykstra算法及其加速方法求解结构约束下的Hankel张量逼近问题,设计交替方向法求解Hankel张量的多重线性低秩逼近问题,数值实验表明新算法是可行的.第四章,研究了二阶张量方程(?)的低秩逼近解.基于Gramian表示,该问题被等价转化为无约束优化问题,构造了求解等价问题的非线性共轭梯度法,数值实验表明新方法比传统的LR-ADI方法和krylov子空间方法收敛速度更快.
[Abstract]:Constrained Zhang Liang approximation is an important subject in the field of numerical algebra. It is widely used in blind source separation, high order statistics, machine learning and harmonic recovery. In this paper, the theory and numerical algorithm of several constrained Zhang Liang approximation problems are studied systematically. In chapter 2, we study the problem of symmetric Zhang Liang multiplex linear low rank approximation (?) We first reconstitute the problem of maximization on Riemannian manifolds, and then extend the spectral conjugate gradient method in Euclidean space to Riemannian manifolds. The spectral conjugate gradient method (RSCG),) on Riemannian manifolds is designed. Then the RSCG method is used to solve the equivalent maximization problem. Finally, the feasibility and validity of the new method are verified by numerical examples. In chapter 3, we study the problem of Hankel Zhang Liang approximation (?) The strong Hankel Zhang Liang approximation problem is transformed into an unconstrained optimization problem by using the van der Mon decomposition of the positive semidefinite Hankel matrix, and the nonlinear conjugate gradient method is used to solve the problem. We design the Dykstra algorithm and its acceleration method to solve the Hankel Zhang Liang approximation problem under structural constraints, and design the alternating direction method to solve the multiplex linear low rank approximation problem of Hankel Zhang Liang. Numerical experiments show that the new algorithm is feasible. In chapter 4, we study the second order Zhang Liang equation (?) Lower rank approximation solution. Based on the Gramian representation, the problem is transformed into an unconstrained optimization problem. A nonlinear conjugate gradient method for solving the equivalent problem is constructed. Numerical experiments show that the new method converges faster than the traditional LR-ADI method and the krylov subspace method.
【学位授予单位】：桂林电子科技大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O224
本文编号：2144118
[Abstract]:Constrained Zhang Liang approximation is an important subject in the field of numerical algebra. It is widely used in blind source separation, high order statistics, machine learning and harmonic recovery. In this paper, the theory and numerical algorithm of several constrained Zhang Liang approximation problems are studied systematically. In chapter 2, we study the problem of symmetric Zhang Liang multiplex linear low rank approximation (?) We first reconstitute the problem of maximization on Riemannian manifolds, and then extend the spectral conjugate gradient method in Euclidean space to Riemannian manifolds. The spectral conjugate gradient method (RSCG),) on Riemannian manifolds is designed. Then the RSCG method is used to solve the equivalent maximization problem. Finally, the feasibility and validity of the new method are verified by numerical examples. In chapter 3, we study the problem of Hankel Zhang Liang approximation (?) The strong Hankel Zhang Liang approximation problem is transformed into an unconstrained optimization problem by using the van der Mon decomposition of the positive semidefinite Hankel matrix, and the nonlinear conjugate gradient method is used to solve the problem. We design the Dykstra algorithm and its acceleration method to solve the Hankel Zhang Liang approximation problem under structural constraints, and design the alternating direction method to solve the multiplex linear low rank approximation problem of Hankel Zhang Liang. Numerical experiments show that the new algorithm is feasible. In chapter 4, we study the second order Zhang Liang equation (?) Lower rank approximation solution. Based on the Gramian representation, the problem is transformed into an unconstrained optimization problem. A nonlinear conjugate gradient method for solving the equivalent problem is constructed. Numerical experiments show that the new method converges faster than the traditional LR-ADI method and the krylov subspace method.
【学位授予单位】：桂林电子科技大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O224
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