IEEE Trans Neural Netw Learn Syst. 2012 Feb;23(2):185-98. doi: 10.1109/TNNLS.2011.2178564.
Recently, many unified learning algorithms have been developed for principal component analysis and minor component analysis. These unified algorithms can be used to extract principal components and, if altered simply by the sign, can also serve as a minor component extractor. This is of practical significance in the implementations of algorithms. This paper proposes a unified self-stabilizing neural network learning algorithm for principal and minor components extraction, and studies the stability of the proposed unified algorithm via the fixed-point analysis method. The proposed unified self-stabilizing algorithm for principal and minor components extraction is extended for tracking the principal subspace (PS) and minor subspace (MS). The averaging differential equation and the energy function associated with the unified algorithm for tracking PS and MS are given. It is shown that the averaging differential equation will globally asymptotically converge to an invariance set, and the corresponding energy function exhibit a unique global minimum attained if and only if its state matrices span the PS or MS of the autocorrelation matrix of a vector data stream. It is concluded that the proposed unified algorithm for tracking PS and MS can efficiently track an orthonormal basis of the PS or MS. Simulations are carried out to further illustrate the theoretical results achieved.
最近，已经开发出许多用于主成分分析和次要成分分析的统一学习算法。这些统一算法可用于提取主成分，如果只是通过符号改变，也可以用作次要成分提取器。这在算法的实现中具有实际意义。本文提出了一种用于提取主成分和次要成分的统一自稳定神经网络学习算法，并通过定点分析方法研究了所提出的统一算法的稳定性。所提出的用于提取主成分和次要成分的统一自稳定算法被扩展用于跟踪主子空间（PS）和次要子空间（MS）。给出了用于跟踪 PS 和 MS 的统一算法的平均微分方程和相关能量函数。结果表明，平均微分方程将全局渐近收敛到不变集，并且相应的能量函数表现出唯一的全局最小值，当且仅当其状态矩阵跨越自相关矩阵的 PS 或 MS 或向量数据流时。得出的结论是，所提出的用于跟踪 PS 和 MS 的统一算法可以有效地跟踪 PS 或 MS 的正交基。进行了模拟以进一步说明所获得的理论结果。
