Noise-bias decomposition for linear regularization

For a bounded linear regularization operator $R_{\alpha }$ and noisy data with $\Vert f_{\delta }-f\Vert \le \delta$, the triangle inequality givesThe first term is noise amplification; the second is the approximation bias on exact data. Thus $\alpha (\delta )\to 0$ and $\delta \Vert R_{\alpha (\delta )}\Vert \to 0$ suffice for a convergent regularization of an inverse problem, provided $R_{\alpha }$ is consistent on exact data.