不同的数据预处理对L1距离性能的影响

问题由来

这个问题来自于CS231n Assignment1 Q1 inlineQuestion 2 ，原问题描述如下
We can also use other distance metrics such as L1 distance.
For pixel values $p_{ij}^{(k)}$ at location $(i,j)$ of some image $I_k$,

the mean $\mu$ across all pixels over all images is $$\mu =\frac{1}{nhw}\sum _{k=1}^n\sum _{i=1}^h\sum _{j=1}^w{p}_{ij}^{(k)}$$
And the pixel-wise mean ${\mu }_{ij}$ across all images is
$${\mu }_{ij}=\frac{1}{n}\sum _{k=1}^n{p}_{ij}^{(k)}.$$
The general standard deviation $\sigma$ and pixel-wise standard deviation ${\sigma }_{ij}$ is defined similarly.

Which of the following preprocessing steps will not change the performance of a Nearest Neighbor classifier that uses L1 distance? Select all that apply.

1. Subtracting the mean $\mu$ (${\tilde{p}}_{ij}^{(k)}=p_{ij}^{(k)}-\mu$.)
2. Subtracting the per pixel mean ${\mu }_{ij}$ (${\tilde{p}}_{ij}^{(k)}=p_{ij}^{(k)}-{\mu }_{ij}$.)
3. Subtracting the mean $\mu$ and dividing by the standard deviation $\sigma$.
4. Subtracting the pixel-wise mean ${\mu }_{ij}$ and dividing by the pixel-wise standard deviation ${\sigma }_{ij}$.
5. Rotating the coordinate axes of the data.

我的思路是这样的：
使用L1 distance，P1=(x1,x2)P_1 = (x_1,x_2)P1​=(x1​,x