The Proximity Operator of a Function with Multiple Affine Mapping Let $f(\mathbf{x}) = g(\mathbf{A}\mathbf{x})$, where $\mathbf{A} \in \mathbb{R}^{M \times N}$ is a linear transformation satisfying $\mathbf{A}\mathbf{A}^T = \mathbf{I}$. Then for any $\mathbf{x} \in \mathbb{R}^{N}$,
\begin{equation}
\text{prox}_f (\mathbf{x}) = \mathbf{x} + \mathbf{A}^T (\text{prox}_g(\mathbf{Ax}) − \mathbf{Ax}).
\end{equation}
Now, if $f(\mathbf{x}) = \sum_{p=1}^{P} g(\mathbf{A}_p\mathbf{x})$, where $\mathbf{A}_p \in \mathbb{R}^{M \times N}$ are multiple linear trasformations satisfying $\mathbf{A}_p\mathbf{A}_p^T = \mathbf{I}$. Then for any $\mathbf{x} \in \mathbb{R}^{N}$, what would be the proximal mapping for the new $f(\mathbf{x})$?.

Even in the simpler case where $P=2$ and $A_1=A_2= \textbf{I}$, there does not exist a closed form solution for the proximal operator of the sum. If you are interested in solving an optimization problem check the keywords "Douglas-Rachford" and "splitting".
