It has been shown that the squared Euclidean distance between two z-normalized vectors x and y each of length m can be written as:
Where mu and sigma are the mean and standard deviation, respectively. I need to derive the above relationship but I am getting stuck and would appreciate any help on where I might be going wrong. Below is my closest attempt:
The issue occurs on the 4th line from the bottom in your original post, since $$ 2m\left[1-\frac{x\cdot y-m\mu_y\sum\frac{x}{m}}{m\sigma_x\sigma_y}\right] \neq 2m\left[1-\frac{x\cdot y-m\mu_y\sum\mu_x}{m\sigma_x\sigma_y}\right] $$ The difference is $$ \frac{x}{m}\neq \mu_x $$ but $$ \sum_{\text{all x}} \frac{x}{m} = \mu_x $$ so putting this in $$ 2m\left[1-\frac{x\cdot y-m\mu_y\sum\frac{x}{m}}{m\sigma_x\sigma_y}\right] = 2m\left[1-\frac{x\cdot y-m\mu_y\mu_x}{m\sigma_x\sigma_y}\right] $$ Which allows for the proof to continue.