ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedverified 2026-10-22 (gpt-6.1-sol)

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Exact truncated Landweber and a learned finite-depth update

Example

Assume Countable Choice. Let $X = Y = R$ , let $A x = a x$ for a fixed $a > 0$ , and let the exact datum be $y = a x^{†}$ with $x^{†} \neq 0$ and noise bound $δ = 0$ . Choose a discrepancy factor $τ > 1$ , $N \geq 1$ , and $0 < ω a^{2} < 1$ . The exact Landweber prefix starts at $x_{0} = 0$ and uses $x_{k + 1} = x_{k} + ω a (y - a x_{k})$ through its finite budget $N$ . For a learned finite-depth architecture with zero initialization, identity update blocks, and layer parameters ${\tilde{ω}}_{0}, \dots, {\tilde{ω}}_{N - 1}$ , the updates are ${\tilde{x}}_{k + 1} = {\tilde{x}}_{k} + {\tilde{ω}}_{k} a (y - a {\tilde{x}}_{k})$ . If every ${\tilde{ω}}_{k} = ω$ , the two depth- $N$ outputs coincide; otherwise the learned output is the finite composition determined by its own layer parameters.

Example references

The exact Landweber recurrence and finite-budget status.

Clason, full lecture notes, Chapter 7

Gregor and LeCun, fixed-depth learned sparse-code predictor

Monga, Li, and Eldar, finite algorithm-unrolling cascades

The layer-parameter and data-consistency conventions.

Facts & Assumptions

Given: Countable Choice, $a > 0$ , $x^{†} \neq 0$ , $y = a x^{†}$ , noise bound $δ = 0$ , discrepancy factor $τ > 1$ , $N \geq 1$ , $0 < ω a^{2} < 1$ , and the stated scalar exact and learned recurrences.

[F1]

Landweber iteration with discrepancy and budget statuses: under Countable Choice, the path starts at $x_{0} = 0$ , uses $x_{k + 1} = x_{k} + ω A^{*} (y - A x_{k})$ , and, with supplied discrepancy factor $τ > 1$ , the finite procedure returns at its first residual test $∥ y - A x_{k} ∥ \leq τ δ$ or at budget $N$ .

[F2]

Unrolled regularization architecture: the deployed unrolled map is a finite composition of its declared update maps with fixed layer parameters.

[F3]

A continuous map has Borel preimages of Borel sets: a continuous map between topological spaces has Borel preimages of Borel sets.

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): the Countable Choice axiom is assumed for the published Landweber interface; the scalar recurrence calculation below makes no additional choice.

Verification

technique · direct

1.1A1F1F2F3givenalgebra

The scalar map $A : R \to R$ , $x \mapsto a x$ , is nonzero and compact because $a > 0$ and the space is finite-dimensional. For every $x$ , $∥ A x ∥_{2} = a ∣ x ∣$ , so $∥ A ∥ = a$ . For every $u, v \in R$ , $⟨ A u, v ⟩ = a u v = ⟨ u, a v ⟩$ , hence $A^{*}$ is also multiplication by $a$ . With [A1], $0 < ω a^{2} < 1$ implies the supplier's condition $0 < ω < 2 / ∥ A ∥^{2}$ . The initialization and scalar update formulas are continuous, hence Borel by [F3], so they satisfy [F2]'s estimator-domain condition. Together with the supplied $τ > 1$ and noise bound $δ = 0$ , these verify every input condition of [F1], which gives the exact recurrence and its finite-budget stopping rule for this instance.

1.2F1givenalgebra

Substituting $y = a x^{†}$ gives $x_{k + 1} - x^{†} = (1 - ω a^{2}) (x_{k} - x^{†})$ . Starting from $x_{0} = 0$ , induction gives $x_{k} = x^{†} [1 - (1 - ω a^{2})^{k}]$ for every $0 \leq k \leq N$ , in particular $x_{N} = x^{†} [1 - (1 - ω a^{2})^{N}]$ . Hence $∣ y - a x_{k} ∣ = a ∣ x^{†} ∣ (1 - ω a^{2})^{k} > 0$ for every such finite $k$ . Since $τ δ = 0$ , the discrepancy test fails even at $k = N$ , so no earlier discrepancy stop occurs and the procedure returns this value at budget exhaustion.

2.1F2F3givenstep 1.2algebra∎

For the learned architecture, ${\tilde{x}}_{0} = 0$ and its displayed recurrence give ${\tilde{x}}_{k + 1} - x^{†} = (1 - {\tilde{ω}}_{k} a^{2}) ({\tilde{x}}_{k} - x^{†})$ . Starting with ${\tilde{x}}_{0} - x^{†} = - x^{†}$ , repeated substitution yields ${\tilde{x}}_{N} - x^{†} = - x^{†} \prod_{k = 0}^{N - 1} (1 - {\tilde{ω}}_{k} a^{2})$ , or ${\tilde{x}}_{N} = x^{†} [1 - \prod_{k = 0}^{N - 1} (1 - {\tilde{ω}}_{k} a^{2})]$ . The fixed-parameter updates are continuous and Borel by [F3], so this is an admissible finite point map. If all layer parameters equal $ω$ , the formula is the exact Landweber value in step 1.2; for other parameters it is the finite composition specified by [F2].

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Sources

Vishal Monga, Yuelong Li, and Yonina C. Eldar, Algorithm Unrolling: Interpretable, Efficient Deep Learning for Signal and Image Processing (standard reference, not scraped)
Christian Clason, Regularization of Inverse Problems, full lecture notes (standard reference, not scraped)
Karol Gregor and Yann LeCun, Learning Fast Approximations of Sparse Coding (standard reference, not scraped)