【文献阅读】A multilayer RBF network and its supervised learning

（1）传统RBF与多层RBF的对比

Since RBF networks use only fixed basis functions, even suppose their centers and covariances can be adjusted, their representation power could be largely restricted.
e.g. it will take a large number of basis functions with small covariances and close centers to approximate a complicated function.
When no a priori knowledge is available on the target input-output function and distribution of input signals, one has to prepare the basis functions with various centers and variances. To estimate these parameters by other methods proved also difficult.

RBF网络使用固定的基函数，只有中心节点与协方差可调整，其泛化能力受限。如果用传统RBF近似一个复杂的函数，会用到许多基函数，且基函数具有小协方差和相近的中心节点。当没有关于目标输入输出函数和输入信号分布的先验知识时，就必须准备具有不同中心和方差的基函数。用其他方法来估计这些参数也被证明是困难的。

If we extend the RBF network to multilayer one, it will have certainly higher representation capability than the original multilayer networks since the prototype activation functions are taken as RBF basis. On the other hand, the local minima problem could be more serious since now the activation functions at each hidden layers are not monotonous functions as sigmoid any longer.

如果将RBF网络扩展到多层网络，它肯定会有比原始多层网络更高的泛化能力，因为此类情况下，原始多层网络的激活函数将作为RBF的基函数之一。但另一方面，局部最小问题可能更严重，因为现在每个隐藏层的激活函数不再是类似sigmoid的单调函数。

（2）文章工作

This paper is to define multilayer RBF network and provide it with global convergent training algorithm. A generalized error back propagation algorithm is obtained for the multilayer RBF network. Then the so-called magic-brush algorithm is applied based on the gradient information derived from the generalized BP algorithm. Issues on extension of the multilayer RBF networks using a pyramid topology are also considered.
Simulations are carried out on hand-written figure recognition to compare the proposed network with the original RBF network and multilayer network with sigmoid activation functions. The multilayer RBF network with the global training algorithm outperformed the other networks in both the output error criterion and the generalization error evaluated by cross validation.

本文定义了多层RBF网络，并为其提供了全局收敛的训练算法。得到了一种针对多层RBF网络的广义误差反向传播算法。然后基于广义BP算法导出的梯度信息，应用所谓的魔刷算法。并讨论了利用金字塔拓扑扩展多层RBF网络的问题。
对手写图形识别进行了仿真，将该网络与原始RBF网络、具有sigmoid激活函数的多层网络进行了比较。采用全局训练算法的多层RBF网络在输出误差准则和交叉验证评估的泛化误差方面均优于其他网络。