Finding the gradient of a norm in a minimization problem

Question

I have to find the gradient of the following term with respect to $X_{1}$:

$\|\Phi\circ(X_{1}-X_{2})-u\|_F^2$ ,

where $u\in\mathbb{R}^{n}$; $X_{1}, X_{2}\in\mathbb{R}^{N\times J}$ and $\Phi\in\mathbb{R}^{n\times NJ}$. The $\circ$ operation is defined as follows:

$\Phi\circ X=\Sigma_{i=1}^{J}\Phi_{i} x_{i}$,

where $x_i$ are the columns of $X$ and $\Phi_{i}$ are $n\times N$ submatrices of $\Phi$.

I know that $\frac{\partial}{\partial X}\|X\|_{F}^{2}=2X$, but the $\circ$ operation is making it rather complicated. Can somebody help?

P.S. Also, would changing the Frobenius norm to l2 norm make any difference here (since $(\Phi\circ(X_{1}-X_{2})-u)$ is a column vector)?

Answer 1

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 $\Phi^{i}$.