Tikhonov regularization and filter functions

Minimize one half of the squared residual plus α/2 times the squared object norm. The factor one half fixes the parameter convention: differentiation gives (A*A+αI)x=A*yδ. The operator on the left is bounded, self-adjoint, and bounded below by α, hence invertible for every α>0. The objective is strongly convex and has a unique minimizer for every datum, even when the original equation has no exact solution or many solutions. This is a property of the regularized problem, not proof that the original inverse became identifiable.

The penalty favors the zero-centered minimum-norm solution. A reference object would replace x by x−x₀, changing the selected nullspace component. A general derivative or smoothing penalty has a different nullspace and needs its own coercivity conditions. In particular, one cannot assume uniqueness from the word quadratic if the data and penalty share an uncontrolled null direction. The identity penalty avoids that issue and allows a clean spectral calculation before more structured penalties are introduced.

In a singular direction with value σ, the data-to-object multiplier is σ/(σ²+α), rather than 1/σ. Its maximum over nonnegative σ is 1/(2√α), reached at σ=√α. Thus fixed positive α bounds noise amplification. When the input datum is exact, the corresponding object coefficient is multiplied by σ²/(σ²+α). This latter number is a shrinkage or resolution factor between zero and one. Confusing those two multipliers obscures whether one is discussing noise gain or bias on a true object.

Directions with σ² much larger than α are nearly preserved, while those with σ² much smaller than α are strongly damped. A hard truncated-SVD filter instead drops selected modes abruptly; both methods trade weak-direction bias for stability but produce different reconstructions. Nullspace components remain zero under the zero-centered identity penalty. Increasing α is therefore not additional measurement information: it increases reliance on the preferred object and can erase a real weak-direction feature along with its noise.

For any fixed minimum-norm solution, taking α→0 removes exact-data bias, but a rate needs information about how that object occupies weak singular directions. A standard source condition is x†=(A*A)^νw with 0<ν≤1 and bounded w. Spectrally, multiplication by σ^(2ν) limits those weak coefficients. The residual multiplier α/(σ²+α) then yields bias bounded by a constant times α^ν. This exponent convention uses A*A; writing a power of |A| instead would double the corresponding exponent.

Balancing α^ν against δ/√α gives α proportional to δ^(2/(2ν+1)) when the source bound is treated as fixed, and object error of order δ^(2ν/(2ν+1)). These are conditional deterministic rates, not automatic accuracies for arbitrary images. Classical identity Tikhonov has qualification one in this source convention, so extending the same rate formula to arbitrarily large ν needs further analysis. At a fixed noisy datum, driving α all the way to zero can undo the regularization rather than improve the result.