I have $$ f(x) = \frac{1}{2}\|y - A*x\|_2^2, $$ where x is matrix of size $512 \times 512$ and A is kernel of shape $5 \times 5$ for example. Then I want to find derivative of $f$ by x:
Rewrite this in operator form as: $$ f(x) = \frac{1}{2}\|y - \mathcal{A}x\|_2^2, $$ here $\mathcal{A}$ is convolution operator, so derivative is: $$ \nabla f = - \mathcal{A}^*(y - \mathcal{A}x) $$
but what is $\mathcal{A}^*$?