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The simpler shifted inverse is Lavrentiev regularization. Its expansion in the orthonormal eigenbasis isUsing $y=Ax$, the exact component error isConsequently the noise-bias decomposition for linear regularization givesFor a prior or data-based bound $X\ge \Vert x\Vert$, balancing noise and approximation bias in the last expression givesFor example $X=(\Vert y_{\delta }\Vert +\delta )/{\lambda }_1$ is computable from the data and the stated noise bound. With fixed positive ${\lambda }_1$, $\alpha \to 0$ and $\delta /\alpha \to 0$ are sufficient for the robust bound to converge.

The first bound is stronger in the finite-dimensional setting. If $X>0$, choosing $\alpha =\delta /X$ gives $\Vert x_{\alpha }^{\delta }-x\Vert \le 2\delta /({\lambda }_1+\delta /X)$. The unregularized inverse already has error at most $\delta /{\lambda }_1$; the balancing rule is a useful stabilized choice rather than a necessary cure for an unbounded inverse. A universal noise-only optimal $\alpha$ cannot be inferred without information about the size or spectral content of $x$.