论文标题
在拉格朗日 - 纽顿框架中的耦合主组件分析的学习规则的推导
Derivation of Learning Rules for Coupled Principal Component Analysis in a Lagrange-Newton Framework
论文作者
论文摘要
我们描述了一个Lagrange-Newton框架，用于具有理想的收敛属性的学习规则，并将其应用于主成分分析（PCA）的情况。在此框架中，将牛顿下降应用于扩展变量向量，该向量还包括带有约束的Lagrange乘数。牛顿下降可以保证各个方向相等的收敛速度，但也需要在系统中产生稳定的固定点，并具有扩展的状态向量。该框架生成“耦合”的PCA学习规则，该规则同时估算了交叉耦合微分方程中的特征向量和相应的特征值。我们证明了这种方法对两个PCA学习规则的可行性，一个用于估计本金，另一个用于估计任意特征向量 - 元素值对（EIGENPAIR）。
We describe a Lagrange-Newton framework for the derivation of learning rules with desirable convergence properties and apply it to the case of principal component analysis (PCA). In this framework, a Newton descent is applied to an extended variable vector which also includes Lagrange multipliers introduced with constraints. The Newton descent guarantees equal convergence speed from all directions, but is also required to produce stable fixed points in the system with the extended state vector. The framework produces "coupled" PCA learning rules which simultaneously estimate an eigenvector and the corresponding eigenvalue in cross-coupled differential equations. We demonstrate the feasibility of this approach for two PCA learning rules, one for the estimation of the principal, the other for the estimate of an arbitrary eigenvector-eigenvalue pair (eigenpair).
