python softmax函数

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数学公式

• $\text{对于 } x \in \mathbb{R}^{1\times n} \text{, } softmax(x) = softmax(\begin{bmatrix} x_1 && x_2 && … && x_n \end{bmatrix}) = \begin{bmatrix} \frac{e^{x_1}}{\sum_{j}e^{x_j}} && \frac{e^{x_2}}{\sum_{j}e^{x_j}} && ... && \frac{e^{x_n}}{\sum_{j}e^{x_j}} \end{bmatrix}$

• $\text{对于一个矩阵 } x \in \mathbb{R}^{m \times n} \text{, $x_{ij}$表示$x$的第 $i^{th}$行 与第 $j^{th}$列,我们有: }$

  $$softmax(x) = softmax\begin{bmatrix} x_{11} & x_{12} & x_{13} & \dots & x_{1n} \\ x_{21} & x_{22} & x_{23} & \dots & x_{2n} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ x_{m1} & x_{m2} & x_{m3} & \dots & x_{mn} \end{bmatrix} = \begin{bmatrix} \frac{e^{x_{11}}}{\sum_{j}e^{x_{1j}}} & \frac{e^{x_{12}}}{\sum_{j}e^{x_{1j}}} & \frac{e^{x_{13}}}{\sum_{j}e^{x_{1j}}} & \dots & \frac{e^{x_{1n}}}{\sum_{j}e^{x_{1j}}} \\ \frac{e^{x_{21}}}{\sum_{j}e^{x_{2j}}} & \frac{e^{x_{22}}}{\sum_{j}e^{x_{2j}}} & \frac{e^{x_{23}}}{\sum_{j}e^{x_{2j}}} & \dots & \frac{e^{x_{2n}}}{\sum_{j}e^{x_{2j}}} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ \frac{e^{x_{m1}}}{\sum_{j}e^{x_{mj}}} & \frac{e^{x_{m2}}}{\sum_{j}e^{x_{mj}}} & \frac{e^{x_{m3}}}{\sum_{j}e^{x_{mj}}} & \dots & \frac{e^{x_{mn}}}{\sum_{j}e^{x_{mj}}} \end{bmatrix} = \begin{pmatrix} softmax\text{(first row of x)} \\ softmax\text{(second row of x)} \\ ... \\ softmax\text{(last row of x)} \\ \end{pmatrix}$$

代码

def softmax(x):
    x_exp = np.exp(x)
    #如果是列向量，则axis=0
    x_sum = np.sum(x_exp, axis = 1, keepdims = True)
    s = x_exp / x_sum    
    return s