 Finite termination of Bregman iteration on a singular vector

For a generalized singular vector, exact data $f=\gamma K{u}_{\lambda }$, and fixed $\alpha >0$, a compatible branch of Bregman iteration has $p^k=\min \{k\gamma /\alpha ,\lambda \}{K}^{\ast }K{u}_{\lambda }$ and the displayed $u^k$. Verify these formulas with the subgradient characterization of an absolutely one-homogeneous functional; before activation the dual coefficient remains at most $\lambda$, and after activation it is $\lambda$. The branch reaches $\gamma u_{\lambda }$ at $k_{\ast }=\lceil \alpha \lambda /\gamma \rceil +1$. Every minimizer branch has the same predicted data, since the quadratic data fidelity is strictly convex in $Ku$. Exact recovery of this particular vector for arbitrary branch choices needs uniqueness, for example injectivity of $K$. For $K(x,y)=x$ and $J(x,y)=|x|$, the second coordinate of every minimizer is arbitrary.