# functions

## logistic sigmoid函数

$$
\sigma(x) = \frac{1}{1+\exp(-x)}
$$

![](http://windmissing.github.io/images_for_gitbook/mathematics_basic_for_ML/2.png)

意义：

1. $$\sigma(x) \in (0, 1)$$
2. 通常用来产生Bernoulli分布中的参数$$\phi$$
3. 当|x|非常大时会饱和，饱和是指$$\sigma'(x)$$的变化非常缓慢。

## softplus函数

$$
\zeta(x) = \log(1+\exp(x))
$$

![](http://windmissing.github.io/images_for_gitbook/mathematics_basic_for_ML/3.png)

意义：

1. $$\zeta(x) \in (0, +\infty)$$
2. 用于产生高斯分布的$$\beta$$或$$\sigma$$参数，$$\beta = \frac{1}{\sigma^2}$$
3. 是$$x\_+ = max(0, x)$$函数是平滑形式

有用性质：

![](http://windmissing.github.io/images_for_gitbook/mathematics_basic_for_ML/4.png)

## 径向基函数 Radial Basis Function

将一个点到另一个点的距离映射成一个实值的函数。\
这里面有三个未知：\
（1）另一个点是什么点？默认是原点，也可以是指定点p。\
（2）距离是什么距离？一般都使用欧氏距离\
（3）对距离做怎样的操作？不同的RBF只要是这一点的不同。

### 欧氏径向基

距离为欧氏距离：

$$
\begin{aligned} r(x) = ||x||\_2 \ r(x, p) = ||x-p||\_2 \end{aligned}
$$

操作为线性操作：

$$
\phi(r) = r
$$

### 高斯径向基

距离为欧氏距离\
操作为高斯函数：

$$
\phi(r) = \exp(-\frac{r^2}{2\sigma^2})
$$


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