论文标题
对称对称PCA学习规则的推导从新颖的目标函数中
Derivation of Symmetric PCA Learning Rules from a Novel Objective Function
论文作者
论文摘要
在正常限制下，可以通过最大化目标函数（在子空间轴上的投影的求和差异）来得出主组件 /子空间分析（PCA / PSA）的神经学习规则。对于具有单轴的子空间，优化产生了数据协方差矩阵的主要特征向量。然后，可以使用缩放过程进行分层学习规则来提取多个特征向量。但是，对于具有多个轴的子空间，优化会导致PSA学习规则，该规则仅收敛到跨越主要子空间但不转移到主要特征向量的轴。必须引入具有不同权重因子的修改目标功能，生产PCA学习规则。多个轴的目标函数的优化导致不需要放气过程的对称学习规则。对于PCA案例，根据权重因子的顺序，对估计的主特征向量（W.R.T.相应的特征值）进行排序（W.R.T.）。 在这里，我们介绍了一个替代目标函数，没有必要引入固定重量因子。取而代之的是，替代目标函数使用平方求和。优化导致对称的PCA学习规则，该规则会收敛到主要特征向量，但没有施加命令。代替具有固定重量因子的对角线矩阵，学习规则中出现了可变的对角矩阵。我们通过确定约束优化的固定点来分析这种替代方法。分析了约束目标函数在固定点上的行为，这既证实了PCA行为又证实了未施加秩序的事实。提出了从目标函数的优化中得出学习规则的不同方法。探索了术语在从这些派生中获得的学习规则中的作用。
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.
