1. $\text{tanh}(\cdot)$ 与 Logistic Sigmoid 函数的联系
$$\text{tanh}(z)=\cfrac{e^z-e^{-z}}{e^z+e^{-z}}=2\sigma(2z)-1\tag{1}$$
下面两个线性模型是等价的
$$y(\mathbf{x},\mathbf{w})=w_0+\sum_{j=1}^Mw_j\sigma\left(\cfrac{x-\mu_j}{s}\right)\tag{2}$$ $$y(\mathbf{x},\mathbf{u})=u_0+\sum_{j=1}^Mu_j\text{tanh}\left(\cfrac{x-\mu_j}{2s}\right)\tag{3}$$ 其中系数之间的关系为$$\begin{array}{r,c,l}w_0 &=& u_0 - \sum\limits_{j=1}^M u_j \qquad j=1,…,M\\ w_j&=&2u_j\end{array}\tag{4}$$
参考资料
- Pattern Recognition and Machine Learning (PRML)