I've posted this question on math overflow but got no answer, so I think it might not be a research level question so I decided to post it here too.
Let $A$ be an orthogonal matrix. It is well known that the proximal mapping $prox_{f \circ A}(x)$ can be evaluated efficiently by $$ (I + \partial (f \circ A))^{-1}(x) = prox_{f \circ A}(x) = A^T prox_{f}(A x), $$ as described here:
Proximal mapping for composition of functions
I was wondering if this extends to a proximal mapping with a diagonal matrix $D$ as step size: $$ (I + D \partial (f \circ A))^{-1}(x) = ? $$ I'm asking if the above proximal operator is easily to evaluate, assuming that $(I + D \partial f)^{-1}$ is easy.
I have been trying quite a while to find some closed-form for the above, but it seems difficult. I do not want to use an iterative method like ADMM to evaluate the proximal mapping.