仿射函数的仿射函数还是仿射函数

证明：仿射函数的仿射函数还是仿射函数
$证明：$
$$\begin{array}{r}\begin{array}{rl}\mathbf{H}& =\mathbf{X}{\mathbf{W}}^{(1)}+{\mathbf{b}}^{(1)},\\ \mathbf{O}& =\mathbf{H}{\mathbf{W}}^{(2)}+{\mathbf{b}}^{(2)}.\end{array}\end{array}$$
$$\begin{array}{r}\begin{array}{rl}\mathbf{O}& =(\mathbf{X}{\mathbf{W}}^{(1)}+{\mathbf{b}}^{(1)}){\mathbf{W}}^{(2)}+{\mathbf{b}}^{(2)}\\ & =\mathbf{X}{\mathbf{W}}^{(1)}{\mathbf{W}}^{(2)}+{\mathbf{b}}^{(1)}{\mathbf{W}}^{(2)}+{\mathbf{b}}^{(2)}\\ & =\mathbf{XW}+\mathbf{b}\end{array}\end{array}$$
其中
$$\begin{gathered} \mathbf{W}={\mathbf{W}}^{(1)}{\mathbf{W}}^{(2)} \\ \mathbf{b}={\mathbf{b}}^{(1)}{\mathbf{W}}^{(2)}+{\mathbf{b}}^{(2)} \end{gathered}$$

参考

4.1. 多层感知机