Htykuut

In the following link is discussed the uniform Effect of K-means Clustering:

https://www.springer.com/cda/content/document/cda_downloaddocument/9783642298066-c2.pdf?SGWID=0-0-45-1338325-p174318763

He used the following equation:

$ frac{displaystyle2 d(C_1,C_2)}{displaystyle n_1n_2}=frac{displaystyle d(C_1,C_1)}{displaystyle n_1^2}+frac{displaystyle d(C_2,C_2)}{displaystyle n_2^2}+2||m_1−m_2||^2.$

where

$ d(C_1,C_2)=sum_{x_iin C_1}sum_{x_jin C_2}||x_i−x_j||^2$ with $ |C_1|=n_1$ and $|C_2 |=n_2$.

I tried to prove this, but I do not get this. He uses the fact that

$ d(C_1,C_1)=2(n_1-1)sum_{i=1}^{n_1} ||x_i||^2 -4 sum_{1leq i < j leq n_1}langle x_i,x_jrangle.$

$ d(C_2,C_2)=2(n_2-1)sum_{i=1}^{n_2} ||y_i||^2 -4 sum_{1leq i < j leq n_2}langle y_i,y_jrangle.$

$ d(C_1,C_2)=2n_2sum_{i=1}^{n_1} ||x_i||^2+2n_1sum_{i=1}^{n_2} ||y_i||^2 -4 sum_{1leq ileq n_1}sum_{1leq jleq n_2}langle x_i,y_jrangle.$

And furthermore with

$||m_1−m_2||^2=langle sum_{i=1}^{n_1}x_i/n_1-sum_{j=1}^{n_1}y_j/n_2,sum_{i=1}^{n_1}x_i/n_1-sum_{j=1}^{n_1}y_j/n_2 rangle = frac{1}{displaystyle n_1^2}sum_{i=1}^{n_1} ||x_i||^2 + frac{2}{displaystyle n_1^2}sum_{1leq i < jleq n_1} langle x_i, x_jrangle +frac{1}{displaystyle n_2^2}sum_{i=1}^{n_2} ||y_i||^2 + frac{2}{displaystyle n_2^2}sum_{1leq i < jleq n_2} langle y_i, y_jrangle - frac{2}{n_1n_2} sum_{1leq ileq n_1}sum_{1leq jleq n_2}langle x_i,y_jrangle, $

I obtain that

$ frac{displaystyle d(C_1,C_2)}{displaystyle n_1n_2}=frac{displaystyle d(C_1,C_1)}{displaystyle n_1^2}+frac{displaystyle d(C_2,C_2)}{displaystyle n_2^2}+2||m_1−m_2||^2.$

Where is my error?