Many papers use the NMSE function without ever explicitly defining it. I have always assumed that $$MSE(x,y)=\frac 1N \sum_i (x_i-y_i)^2$$ and $$ NMSE(x,y)=MSE(x,y)/MSE(x,0) = \frac{\| x-y\|_2^2}{\| x\|_2^2}$$ where $y$ is the approximation to $x$. This gives a simple relation between NMSE and relative $\ell^2$ error. An internet search however only shows strange definitions like $$\frac{ \sum_i (x_i-y_i)^2}{N\sum_i (x_i)^2} \quad\text{or} \quad \frac{N \sum_i (x_i-y_i)^2}{\sum_i x_i \sum_i y_i}$$

Is my interpretation not the standard definition?