Novel Adaptive Generalized Eigenvector Estimation Algorithm and Its Characteristic Analysis
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摘要: 为了发展快速稳定的广义特征向量估计算法，该文提出基于神经网络的新型单维广义特征向量估计算法；通过分析该算法的所有平衡点证明了当且仅当神经网络权向量等于最小广义特征值对应的广义特征向量时该算法达到稳定状态；利用确定性离散时间分析方法完成了所提算法的动态特性分析，给出了保证算法收敛的边界条件；通过膨胀技术将单维算法扩展为多维广义特征向量估计算法，该算法可以根据实际需要增加提取广义特征向量的数量。仿真实验表明所提算法具有很好地收敛性，而且收敛速度优于一些现有算法。Abstract: In order to develop fast and stable algorithm for estimating generalized eigenvector, a novel neuron-based algorithm is proposed for extracting the single generalized eigenvector. Through analyzing all of the stationary points, it is proved that the single estimation algorithm reaches the steady state if and only if the weight vector of the neural network is equal to the generalized eigenvector corresponding to the smallest generalized eigenvalue of a matrix pencil. The dynamic analysis of the single estimation algorithm is accomplished by the deterministic discrete time method and some boundary conditions are also obtained to guarantee the algorithm’s convergence. Trough applying the inflation technique, the single generalized eigenvector estimation algorithm is extended into a multiple generalized eigenvector estimation algorithm, and the number of the generalized eigenvectors can be increased according to actual requirement. Simulation experiments results prove that the proposed algorithm has good convergence, and the convergence speed is better than some existing algorithms.
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Key words:
- Generalized eigenvector /
- Stability analysis /
- Dynamic analysis /
- Multiple extraction
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表 1 多维广义特征向量估计步骤
初始化：设置学习因子$\eta $和广义特征值上界$\sigma $，产生初始化权向量${\boldsymbol{w}}(0)$ 迭代计算：令$d = 1,2, \cdots ,r$，其中$r$为需要估计广义特征向量的个数 步骤1：令$d = 1$，利用式(5)估计出第1个广义特征向量${{\boldsymbol{\bar v}}_n}$； 步骤2：令$d = d + 1$，利用式(11)计算出第$d$个广义特征向量${{\boldsymbol{\bar v}}_{n - d + 1}}$ $ {{\boldsymbol{w}}_d}(k + 1) = {{\boldsymbol{w}}_d}(k) + \eta \left[ { - \left( {2{\boldsymbol{w}}_d^{\rm{T}}(k){{\boldsymbol{R}}_x}{{\boldsymbol{w}}_d}(k) - 1} \right){\boldsymbol{R}}_x^{ - 1}{{\boldsymbol{R}}_{d,y}}{{\boldsymbol{w}}_d}(k) + {\boldsymbol{w}}_d^{\rm{T}}(k){{\boldsymbol{R}}_{d,y}}{{\boldsymbol{w}}_d}(k){{\boldsymbol{w}}_d}(k)} \right] $ (11) 其中
$ {{\boldsymbol{R}}_{d,y}} = {{\boldsymbol{R}}_y} + \sigma \displaystyle\sum\limits_{d = 1}^{d - 1} {{{\boldsymbol{R}}_x}{{{\boldsymbol{\bar v}}}_{n - d + 2}}{\boldsymbol{\bar v}}_{n - d + 2}^{\rm{T}}{{\boldsymbol{R}}_x}} $ (12)步骤3：重复步骤2，直到$d = r$。 
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