Finding the gradient of a norm in a minimization problem
This is the result of taking the derivative of $\|\Phi\circ(X_{1}-X_{2})-u\|_F^2$ w.r.t. $X_{1}$:

$[\Phi^{1'}(\Phi\circ(X_{1}-X_{2})-u), ...,\Phi^{J'}(\Phi\circ(X_{1}-X_{2})-u)]$ .

This can be got by first finding the expression for the norm, and then taking the matrix derivative (w.r.t. $X_{1}$) to get an $N \times J$ matrix. This matrix is then represented in terms of the sub matrices $\Phi^{i}$.