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Fix a singular system of a compact operator, with $K{v}_j={\sigma }_j{w}_j$ and $K^{\ast }{w}_j={\sigma }_j{v}_j$, ${\sigma }_j>0$. A spectral regularization method has the formIt annihilates $\mathcal{R}(K{)}^{\perp }$. Standard sufficient spectral filter conditions are boundedness of ${\sup }_{0<\sigma \le \Vert K\Vert }|\sigma g_{\alpha }({\sigma }^2)|$ for each $\alpha$, pointwise $\lambda g_{\alpha }(\lambda )\to 1$ as $\alpha \downarrow 0$ for $\lambda >0$, and a uniform bound on $|\lambda g_{\alpha }(\lambda )|$. These give bounded linear operators and convergence on the domain of the Moore–Penrose inverse of an operator by the Picard criterion and dominated convergence theorem.

For Tikhonov regularization,This can be computed by solving the Tikhonov normal equation, without knowing the singular value decomposition.

For spectral cutoff regularization, takewith value zero at $\lambda =0$. Only singular values ${\sigma }_j\ge \sqrt{\alpha }$ are inverted. Here $g_{\alpha }$ multiplies $\sigma$; conventions calling the full coefficient $\sigma g_{\alpha }({\sigma }^2)$ the spectral filter are equivalent.

Use the singular system of a compact operator convention $A{x}_j={\sigma }_j{y}_j$ and $A^{\ast }{y}_j={\sigma }_j{x}_j$. Starting from zero gives no component in $\ker A$. Write $x(t)={\sum }_j{c}_j(t)x_j$. The coefficient equations areSolving these scalar linear ordinary differential equations givesConsequently asymptotic regularization has the spectral regularization method representationThe component of $f$ orthogonal to the closure of the range of $A$ is annihilated by $A^{\ast }$ and contributes nothing. The scalar filter satisfies $(1-e^{-{\sigma }^2/\alpha })/\sigma \le \min (\sigma /\alpha ,1/\sigma )\le {\alpha }^{-1/2}$, so every fixed-$\alpha$ reconstruction is a bounded linear operator. Its long-time limit is the Moore–Penrose inverse of an operator on admissible data.

A regularization of an inverse problem replaces the unbounded generalized inverse by bounded operators $R_{\alpha }:Y\to X$ satisfying $R_{\alpha }y\to A^{†}y$ as $\alpha \downarrow 0$ on admissible exact data. A spectral regularization method can be writtenwhere $f_{\alpha }(\sigma )\to 1$ for fixed $\sigma >0$ while $f_{\alpha }(\sigma )/\sigma$ remains bounded for each fixed $\alpha$. A uniform bound on $f_{\alpha }$ supplies the usual dominated-convergence argument on exact data.

This defines $f_{\alpha }$ as the dimensionless damping factor. In the alternative convention calling the entire inverse coefficient the spectral filter, that filter is $f_{\alpha }(\sigma )/\sigma$. For data error bounded by $\delta$, a sufficient regularization parameter choice also requires $\delta \Vert R_{\alpha (\delta )}\Vert \to 0$; consistency alone does not control noise.

Repeated substitution in Landweber iteration from $u_0=0$ givesOn the $i$th singular vector, $A^{\ast }A$ has eigenvalue ${\sigma }_i^2$ and $A^{\ast }{y}_i={\sigma }_i{x}_i$. The finite geometric series therefore givesThis is a spectral regularization method with filter

For a compact operator $A:X\to Y$ between Hilbert spaces, a singular system consists of positive singular values and orthonormal families withThe $v_j$ form a basis of $(\ker A{)}^{\perp }$, and the $u_j$ form a basis of the closure of its operator range. The family is finite when $A$ has finite rank. For infinite rank, the singular values can be ordered with ${\sigma }_j\downarrow 0$.

The expansion $Ax={\sum }_j{\sigma }_j⟨x,v_j⟩u_j$ separates channels with different data sensitivity. Inverting a channel divides by ${\sigma }_j$, explaining why an infinite-rank compact operator generally has an unbounded inverse on its range. Spectral regularization methods replace those inverse coefficients by bounded filters.

For the diagonal operator on sequence space $A(x_1,x_2,\dots )=(x_1,x_2/2,x_3/3,\dots )$ on L2 space, the standard coordinate vectors give ${\sigma }_j=1/j$. A small error in a late output coordinate can correspond to a much larger change in the unknown. No infinite sequence tending to zero is required for a finite-rank operator.

This range condition puts the exact solution in the operator range of the adjoint operator. It controls the small-singular value components of the solution and supplies convergence rates for spectral regularization methods. It is the quadratic-penalty case of a source condition in variational regularization.

For a bounded linear operator between Hilbert spaces, the spectral regularization method for quadratic Tikhonov regularization has scalar coefficient $s/(s^2+\alpha )$. Since $(s-\sqrt{\alpha }{)}^2\ge 0$, this coefficient is at most $1/(2\sqrt{\alpha })$. The spectral theorem for normal operators transfers this scalar bound to the operator norm, providing the noise term in the noise-bias decomposition for linear regularization.

For a singular value decomposition $A{v}_j={\sigma }_j{u}_j$, truncated SVD with cutoff $\tau >0$ estimates the inverse action by $x_{\tau }={\sum }_{{\sigma }_j\ge \tau }{\sigma }_j^{-1}⟨b,u_j⟩v_j$. Discarding small singular values limits amplification of data errors; this is a spectral regularization method.

Truncated singular value decomposition retains only singular components above a threshold, or equivalently only a finite leading set of singular vectors.