= Finite termination of Bregman iteration on a singular vector
{title2=$u^k=\min\{\gamma,(k\gamma-\alpha\lambda)_+\}u_\lambda$}
For a <generalized singular vector>, exact data $f=\gamma Ku_\lambda$, and fixed $\alpha>0$, a compatible branch of <Bregman iteration> has $p^k=\min\{k\gamma/\alpha,\lambda\}K^*Ku_\lambda$ 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 $k_*=\lceil\alpha\lambda/\gamma\rceil+1$. 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.
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