幾類約束張量逼近的理論與數(shù)值算法研究
發(fā)布時間：2018-07-25 14:31
【摘要】：約束張量逼近問題是數(shù)值代數(shù)領(lǐng)域研究和探討的重要課題.它在盲源分離、高階統(tǒng)計、機器學(xué)習(xí)和諧波恢復(fù)等領(lǐng)域有著廣泛的應(yīng)用.本文系統(tǒng)地研究了幾類約束張量逼近問題的理論和數(shù)值算法.第二章,研究了對稱張量多重線性低秩逼近問題(?)我們首先將該問題重構(gòu)成黎曼流形上的極大化問題,再將歐式空間上的譜共軛梯度法推廣至黎曼流形上,設(shè)計了黎曼流形上的譜共軛梯度法(RSCG),接著利用RSCG方法求解等價的極大化問題,最后用數(shù)值例子驗證了新方法的可行性和有效性。第三章,研究了Hankel張量逼近問題(?)利用半正定Hankel矩陣的范德蒙分解將強Hankel張量逼近問題轉(zhuǎn)化為無約束優(yōu)化問題,并借助非線性共軛梯度法進(jìn)行求解.我們設(shè)計Dykstra算法及其加速方法求解結(jié)構(gòu)約束下的Hankel張量逼近問題,設(shè)計交替方向法求解Hankel張量的多重線性低秩逼近問題,數(shù)值實驗表明新算法是可行的.第四章,研究了二階張量方程(?)的低秩逼近解.基于Gramian表示,該問題被等價轉(zhuǎn)化為無約束優(yōu)化問題,構(gòu)造了求解等價問題的非線性共軛梯度法,數(shù)值實驗表明新方法比傳統(tǒng)的LR-ADI方法和krylov子空間方法收斂速度更快.
[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.
【學(xué)位授予單位】：桂林電子科技大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2017
【分類號】：O224
本文編號：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.
【學(xué)位授予單位】：桂林電子科技大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2017
【分類號】：O224
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