常见的距离函数总结

闵可夫斯基距离：给定样本 $\overrightarrow{\mathbf{x}}_{i}=\left(x_{i, 1}, x_{i, 2}, \cdots, x_{i, n}\right)^{T}$ , $\overrightarrow{\mathbf{x}}_{j}=\left(x_{j, 1}, x_{j, 2}, \cdots, x_{j, n}\right)^{T}$ ，则闵可夫斯基距离定义为：
$\text { distance( }\left(\overrightarrow{\mathbf{x}}_{i}, \overrightarrow{\mathbf{x}}_{j}\right)=\left(\sum_{d=1}^{n}\left|x_{i, d}-x_{j, d}\right|^{p}\right)^{1 / p}$
当p=2时，闵可夫斯基距离就是欧式距离：
$\operatorname{distance}\left(\overrightarrow{\mathbf{x}}_{i}, \overrightarrow{\mathbf{x}}_{j}\right)=\left\|\overrightarrow{\mathbf{x}}_{i}-\overrightarrow{\mathbf{x}}_{j}\right\|_{2}=\sqrt{\sum_{d=1}^{n}\left|x_{i, d}-x_{j, d}\right|^{2}}$
当p=1时，闵可夫斯基距离就是曼哈顿距离：
$\text { distance }\left(\overrightarrow{\mathbf{x}}_{i}, \overrightarrow{\mathbf{x}}_{j}\right)=\left\|\overrightarrow{\mathbf{x}}_{i}-\overrightarrow{\mathbf{x}}_{j}\right\|_{1}=\sum_{d=1}^{n}\left|x_{i, d}-x_{j, d}\right|$