Feng Da-Zheng, Zheng Wei-Xing, Jia Ying
National Laboratory for Radar Signal Processing, Xidian University, 710071 Xi'an, PR China.
IEEE Trans Neural Netw. 2005 May;16(3):513-21. doi: 10.1109/TNN.2005.844854.
A novel random-gradient-based algorithm is developed for online tracking the minor component (MC) associated with the smallest eigenvalue of the autocorrelation matrix of the input vector sequence. The five available learning algorithms for tracking one MC are extended to those for tracking multiple MCs or the minor subspace (MS). In order to overcome the dynamical divergence properties of some available random-gradient-based algorithms, we propose a modification of the Oja-type algorithms, called OJAm, which can work satisfactorily. The averaging differential equation and the energy function associated with the OJAm are given. It is shown that the averaging differential equation will globally asymptotically converge to an invariance set. The corresponding energy or Lyapunov functions exhibit a unique global minimum attained if and only if its state matrices span the MS of the autocorrelation matrix of a vector data stream. The other stationary points are saddle (unstable) points. The globally convergence of OJAm is also studied. The OJAm provides an efficient online learning for tracking the MS. It can track an orthonormal basis of the MS while the other five available algorithms cannot track any orthonormal basis of the MS. The performances of the relative algorithms are shown via computer simulations.
一种基于随机梯度的新型算法被开发出来，用于在线跟踪与输入向量序列自相关矩阵最小特征值相关的次要分量（MC）。用于跟踪单个MC的五种可用学习算法被扩展到用于跟踪多个MC或次要子空间（MS）的算法。为了克服一些基于随机梯度的可用算法的动态发散特性，我们提出了一种对Oja型算法的改进，称为OJAm，它可以令人满意地工作。给出了与OJAm相关的平均微分方程和能量函数。结果表明，平均微分方程将全局渐近收敛到一个不变集。相应的能量或李雅普诺夫函数当且仅当其状态矩阵跨越向量数据流自相关矩阵的MS时呈现唯一的全局最小值。其他驻点是鞍点（不稳定点）。还研究了OJAm的全局收敛性。OJAm为跟踪MS提供了一种有效的在线学习方法。它可以跟踪MS的正交基，而其他五种可用算法无法跟踪MS的任何正交基。通过计算机模拟展示了相关算法的性能。
