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An error bound $A\delta /\alpha +Bc\alpha$, with positive $A,B,c$, is minimized at ${\alpha }_{\ast }=\sqrt{A\delta /(Bc)}$, with value $2\sqrt{ABc\delta }$. Differentiate the scalar bound, or apply the arithmetic-geometric mean inequality. This is an a priori regularization parameter choice for a linear regularization with inverse-parameter noise amplification and first-order approximation bias; the parameter must also lie in the method's allowed interval.

On $(0,1/2)$ use the forward difference and on $(1/2,1)$ the backward difference, with $0<h<1/2$. This defines a bounded linear operator on $L^2(0,1)$ and $\Vert D_h\Vert \le \sqrt{6}/h$. For $f$ in the range of the Volterra integration operator, these differences are local averages of $f^{\prime }$ and converge to $f^{\prime }$ in $L^2$ by continuity of translations. Hence $D_h$ gives a linear regularization of differentiation. No pointwise evaluation of an arbitrary $L^2$ equivalence class is needed: translated functions are defined almost everywhere.

A linear regularization is a family of bounded operators $R_{\alpha }:Y\to X$ such that $R_{\alpha }y\to A^{†}y$ for every $y$ in the domain of the Moore–Penrose inverse of an operator as $\alpha \downarrow 0$. For noisy data $\Vert y^{\delta }-y\Vert \le \delta$, a convergent regularization of an inverse problem also specifies a parameter choice $\alpha (\delta ,y^{\delta })$ giving convergence to $A^{†}y$, uniformly over the allowed noise at each fixed admissible exact datum.

Expand $x_{\alpha }$ in the input singular vectors. The Tikhonov normal equation impliesThe component in $\ker A$ is zero because $\alpha >0$. Comparing with the given filter convention givesThe singular-system Tikhonov filter is often described by its gain $g_{\alpha }(\sigma )=\sigma /(\alpha +{\sigma }^2)$; the requested $f_{\alpha }$ includes one extra factor of $\sigma$. For admissible data, $f_{\alpha }\to 1$ and $0\le f_{\alpha }\le 1$, so dominated convergence theorem applied to the squared Picard criterion coefficients proves exact-data convergence. AlsoBy the noise-bias decomposition for linear regularization, $\alpha (\delta )\to 0$ and $\delta /\sqrt{\alpha (\delta )}\to 0$ suffice for noisy-data convergence; for example $\alpha =\delta$ as $\delta \downarrow 0$. This establishes both consistency and stability of Tikhonov regularization.

A linear regularization is a family of bounded linear operators $R_{\alpha }:V\to U$, $\alpha >0$, for which $R_{\alpha }f\to K^{†}f$ for every $f\in \mathcal{D}(K^{†})$ as $\alpha \downarrow 0$. To obtain a convergent regularization of an inverse problem for noisy data, choose a regularization parameter so that the exact-data approximation error and the amplified noise both vanish.

An example is Tikhonov regularization:The positive quadratic term makes $K^{\ast }K+\alpha I$ invertible. The Tikhonov filter norm bound follows from the filter $s/(s^2+\alpha )$, whose maximum for $s\ge 0$ is $1/(2\sqrt{\alpha })$. Therefore $\Vert R_{\alpha }\Vert \le 1/(2\sqrt{\alpha })$. The spectral factors $s^2/(s^2+\alpha )$ tend to one on the positive spectrum; dominated convergence theorem gives consistency on $\mathcal{D}(K^{†})$. The noise-bias decomposition for linear regularization then proves convergence whenever $\alpha (\delta )\to 0$ and $\delta /\sqrt{\alpha (\delta )}\to 0$, for example $\alpha (\delta )=\delta$ for sufficiently small positive noise levels.

A regularization of an inverse problem is a family of continuous maps $R_{\alpha }:V\to U$, $\alpha >0$, such that $R_{\alpha }f\to K^{†}f$ for each $f\in \mathcal{D}(K^{†})$ as $\alpha \downarrow 0$. A linear regularization requires each $R_{\alpha }$ to be a bounded linear operator.

A regularization parameter choice assigns $\alpha =\alpha (\delta ,f^{\delta })>0$ from the noise bound and possibly the measured data. A convergent regularization of an inverse problem satisfies, for every fixed admissible exact datum,In the usual definition the chosen parameter also tends to zero uniformly over this noise ball. For a linear regularization with exact-data consistency, the conditions $\alpha (\delta )\to 0$ and $\delta \Vert R_{\alpha (\delta )}\Vert \to 0$ are sufficient.

For example, Tikhonov regularization with identity penalty has $R_{\alpha }=(K^{\ast }K+\alpha I{)}^{-1}{K}^{\ast }$ and $\Vert R_{\alpha }\Vert \le 1/(2\sqrt{\alpha })$. Its spectral filter converges to the Moore–Penrose inverse of an operator on its domain. Taking $\alpha (\delta )=\delta$ for small positive noise levels gives both required limits.

For a linear regularization, add and subtract the exact-data reconstruction:The triangle inequality gives the noise-bias decomposition for linear regularization:The two summands are noise amplification and approximation bias. Increasing the strength of regularization typically decreases noise amplification and increases bias; reducing regularization has the opposite effect. These are qualitative tendencies and bounds, not a claim that each realized noise error is monotone for every noise vector.

For the one-sided differentiation example, the bounds are $\sqrt{6}\delta /h$ and $\Vert f^{\prime \prime }{\Vert }_{\infty }h/2$. Their sum has an interior balance point when the second-derivative bound is positive and the optimum lies in the permitted interval. The following original sketch plots these bounds against $h$:

Regularization replaces sensitive inversion by a controlled approximation. For a bounded linear operator between Hilbert spaces, a linear regularization is a family of bounded linear operators $R_{\alpha }$ satisfying $R_{\alpha }y\to A^{†}y$ as $\alpha \downarrow 0$ for each datum in the domain of the Moore–Penrose inverse of an operator. Noisy-data convergence additionally requires a suitable regularization parameter choice.^[ref] Nonlinear variational regularization can instead use nonquadratic penalties.

The noise-bias decomposition for linear regularization separates two costs: amplifying measurement noise and changing the exact-data reconstruction. Truncated singular value decomposition discards weak modes; Tikhonov regularization attenuates them smoothly. A quadratic norm penalty expresses a preference for smaller reconstructions. Neither method manufactures a missing null space component: with complete averaging, the minimum-norm least-squares solution for $(0.5,0.5{)}^T$ is $(0.5,0.5{)}^T$, even if the original image was $(1,0{)}^T$. The preference selects a representative; extra observations or justified constraints are needed to identify the original.

For Tikhonov regularization, the Tikhonov stability bound limits transmitted measurement noise of norm at most $\delta$ to $\delta /(2\sqrt{\alpha })$. Taking $\alpha \to 0$ while $\delta /\sqrt{\alpha }\to 0$ gives a sufficient rule for convergent regularization of an inverse problem. Holding $\alpha$ fixed leaves approximation bias; letting it vanish too quickly can admit overwhelming measurement noise.