With the above linear dependence we can in turn define our optimization problem in terms of the optimization of the mean-squared error, that is we wish to optimize
$$ \min_{\boldsymbol{W},\boldsymbol{V}\in {\mathbb{R}}}\frac{1}{n}\sum_{i=0}^{n-1}\left(x_i-\tilde{x}_i\right)^2=\frac{1}{n}\vert\vert \boldsymbol{x}-\boldsymbol{V}\boldsymbol{W}\boldsymbol{x}\vert\vert_2^2, $$where we have used the definition of a norm-2 vector, that is
$$ \vert\vert \boldsymbol{x}\vert\vert_2 = \sqrt{\sum_i x_i^2}. $$