基于新目标函数的对称PCA学习规则
Derivation of Symmetric PCA Learning Rules from a Novel Objective Function
摘要：
主成分/子空间分析（PCA / PSA）的神经学习规则可以通过在正交正态性约束下最大化目标函数（子空间轴上的投影的总和方差）来导出。对于具有单轴的子空间，优化产生数据协方差阵的主特征向量。然后，可以使用具有放气过程的分层学习规则来提取多个特征向量。然而，对于具有多个轴的子空间，优化导致PSA学习规则仅收敛于跨越主子空间的轴，而不是主特征向量。引入了一个具有不同权重因子的修正目标函数，以产生PCA学习规则。多轴目标函数的优化导致对称的学习规则，不需要放气过程。对于PCA情况，估计的主特征向量被排序（w.r.t.对应的本征值），这取决于加权因子的阶数。在这里，我们引入了一个替代的目标函数，其中没有必要引入固定的权重因子;相反，替代的目标函数使用平方和。优化导致对称PCA学习规则收敛到主特征向量，但没有强加的顺序。代替具有固定权重因子的对角矩阵，可变对角矩阵出现在学习规则中。我们通过确定约束优化的不动点来分析这种替代方法。分析了约束目标函数在不动点处的行为，证实了PCA行为和没有强加顺序的事实。不同的方法来获得学习规则的优化的目标函数。从这些推导获得的学习规则中的条款的作用进行了探讨。
英文摘要：
Neural learning rules for principal component / subspace analysis (PCA / PSA) can be derived by maximizing an objective function (summed variance of the projection on the subspace axes) under an orthonormality constraint. For a subspace with a single axis, the optimization produces the principal eigenvector of the data covariance matrix. Hierarchical learning rules with deflation procedures can then be used to extract multiple eigenvectors. However, for a subspace with multiple axes, the optimization leads to PSA learning rules which only converge to axes spanning the principal subspace but not to the principal eigenvectors. A modified objective function with distinct weight factors had to be introduced produce PCA learning rules. Optimization of the objective function for multiple axes leads to symmetric learning rules which do not require deflation procedures. For the PCA case, the estimated principal eigenvectors are ordered (w.r.t. the corresponding eigenvalues) depending on the order of the weight factors. Here we introduce an alternative objective function where it is not necessary to introduce fixed weight factors; instead, the alternative objective function uses squared summands. Optimization leads to symmetric PCA learning rules which converge to the principal eigenvectors, but without imposing an order. In place of the diagonal matrices with fixed weight factors, variable diagonal matrices appear in the learning rules. We analyze this alternative approach by determining the fixed points of the constrained optimization. The behavior of the constrained objective function at the fixed points is analyzed which confirms both the PCA behavior and the fact that no order is imposed. Different ways to derive learning rules from the optimization of the objective function are presented. The role of the terms in the learning rules obtained from these derivations is explored.
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