= Noise-bias decomposition for linear regularization
For a bounded linear regularization operator $R_\alpha$ and noisy data with $\|f_\delta-f\|\leq\delta$, the <triangle inequality> gives
$$
\|R_\alpha f_\delta-A^\dagger f\|
\leq\delta\|R_\alpha\|+\|R_\alpha f-A^\dagger f\|.
$$
The first term is noise amplification; the second is the approximation bias on exact data. Thus $\alpha(\delta)\to0$ and $\delta\|R_{\alpha(\delta)}\|\to0$ suffice for a <convergent regularization of an inverse problem>, provided $R_\alpha$ is consistent on exact data.
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