Global gradient-norm projection residual ($u=g-P_{A^{\ast }{B}_{\alpha }}g$)

Let $C=A^{\ast }{B}_{\alpha }$ with one global dual ball. Its support function is $\alpha \Vert Au{\Vert }_2$, and the proximal operator of a support function gives $u=g-P_{C}g$. The removed component is the convex projection, while the reconstructed image signal is its residual. Projected gradient descent on $\frac{1}{2}\Vert g-A^{\ast }p{\Vert }_2^2$, $p\in B_{\alpha }$, converges for $0<\tau <2/\Vert A{\Vert }^2$.