= Regularization of an inverse problem
{title2=$x_\alpha^\delta=R_\alpha y^\delta$}
= Inverse-problem regularization
{synonym}
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 $R_\alpha y\to A^\dagger y$ as $\alpha\downarrow0$ for each datum in the domain of the <Moore–Penrose inverse of an operator>. Noisy-data convergence additionally requires a suitable <regularization parameter choice>.https://arxiv.org/pdf/2001.00617[Christian Clason, Regularization of Inverse Problems, §4.1]{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 $\alpha\to0$ while $\delta/\sqrt\alpha\to0$ 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>.
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