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Compact-operator Tikhonov bias vanishes on the minimum-norm solution

Statement

Assume Countable Choice. Let $A : X \to Y$ be compact between real or complex Hilbert spaces, let $x^{†} ⊥ \ker A$ , and put $y = A x^{†}$ . For $λ > 0$ , let $x_{λ} = (A^{*} A + λ I_{X})^{- 1} A^{*} y$ . Then $x^{†} - x_{λ} = \sum_{t \in Σ} \frac{λ}{t + λ} P_{t} x^{†},$ where $Σ$ and $P_{t}$ are the nonzero eigenvalues and eigenspace projections of $A^{*} A$ , and $\lim_{λ ↓ 0} ∥ x^{†} - x_{λ} ∥ = 0.$

Facts & Assumptions

Given: Countable Choice, compact $A : X \to Y$ between Hilbert spaces, $x^{†} ⊥ \ker A$ , $y = A x^{†}$ , and $λ > 0$ .

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): Countable Choice is the hypothesis of the compact self-adjoint spectral theorem used below.

[F1]

A compact Hilbert-space Gram operator is positive and has the forward kernel: $S = A^{*} A$ is compact, positive and self-adjoint, with $\ker S = \ker A$ .

[F2]

Spectral theorem for compact self adjoint operators: For compact self-adjoint $S$ , the nonzero eigenspaces are orthogonal and their closed span is $(\ker S)^{⊥}$ ; finite-subset spectral sums converge in norm.

[F3]

Compact spectral filters, source conditions, and qualification: $x_{λ} = (A^{*} A + λ I)^{- 1} A^{*} y$ uses the multiplier $g_{λ} (t) = (t + λ)^{- 1}$ .

[F4]

The shifted Hilbert-space normal operator is invertible with inverse norm at most $1 / λ$ : For bounded $A$ and $λ > 0$ , $A^{*} A + λ I$ is invertible.

[F5]

Orthogonality and the orthogonal complement: A vector in $(\ker S)^{⊥}$ is orthogonal to every vector in $\ker S$ .

Proof

1.1A1F1F2F5

By [F1], $\ker S = \ker A$ . Thus $x^{†} ⊥ \ker S$ , and the spectral decomposition in [F2] gives $x^{†} = \sum_{t \in Σ} P_{t} x^{†}$ as a norm-convergent finite-subset net.

2.1F1F3F4step 1.1algebra

Since $y = A x^{†}$ , $A^{*} y = S x^{†}$ . The filter in [F3] therefore gives $x_{λ} = g_{λ} (S) S x^{†}$ , whose coefficient on $E_{t}$ is $t / (t + λ)$ . The coefficient on $\ker S$ is zero. Subtracting from the expansion in step 1.1 proves the displayed bias identity.

3.1F2step 2.1algebra∎

Given $ε > 0$ , choose a finite $F \subseteq Σ$ so the norm of the tail $\sum_{t \notin F} P_{t} x^{†}$ is less than $ε / 2$ ; this is possible because the finite-subset net in [F2] converges. Each scalar $λ / (t + λ)$ for $t \in F$ tends to zero, so for all sufficiently small $λ > 0$ the norm of the finite part of the bias sum is less than $ε / 2$ . The tail has norm at most its unfiltered norm because $0 \leq λ / (t + λ) \leq 1$ . The triangle inequality then gives $∥ x^{†} - x_{λ} ∥ < ε$ .

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Sources

Guido Clason, Regularization of Inverse Problems, full lecture notes (standard reference, not scraped)
Per Christian Hansen, Introduction to Inverse Problems, Chapter 4: Regularization Methods (standard reference, not scraped)