ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generated

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Landweber reconstructs a three-pixel blurred signal

Example

Consider the symmetric three-pixel averaging blur $A = \frac{1}{4} (\begin{array}{ccc} 2 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 1 & 2 \end{array}) .$ For the zero-sum signal $x^{†} = (1, 0, - 1)^{T}$ , exact data are $y = A x^{†} = \frac{1}{4} x^{†}$ . With $x_{0} = 0$ and $ω = 1$ , the Landweber iterates are $x_{k} = (1 - (15 / 16)^{k}) x^{†}$ : the first reconstruction is only $x_{1} = \frac{1}{16} x^{†}$ , while the exact signal is recovered in the limit.

Facts & Assumptions

Given: The matrix, signal, exact data, and iteration described in the Example.

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): Countable Choice is assumed by the Hilbert-adjoint supplier used in Landweber iteration.

[F1]

Landweber iteration with discrepancy and budget statuses: Its infinite mathematical path satisfies $x_{k + 1} = x_{k} + ω A^{T} (y - A x_{k})$ for real finite-dimensional data.

[F2]

Landweber iterates and residuals are spectral-filter iterates: $x^{†} - x_{k} = (I - ω A^{T} A)^{k} x^{†}$ for exact data.

[F3]

The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ : The Euclidean inner product is $u^{T} v$ , with norm $∥ u ∥_{2} = (u^{T} u)^{1 / 2}$ .

[F4]

Rectangular matrix multiplication and the identity matrix $I_{n}$ , including zero-sized shapes: Matrix multiplication acts on a vector by the coordinate sums of the rows.

[F5]

The operator norm is zero on the zero domain and otherwise is max_{||v||=1} ||Tv||: The induced operator norm is $\sup_{∥ v ∥_{2} \leq 1} ∥ A v ∥_{2}$ .

Verification

Given: The displayed matrix $A$ , vector $x^{†}$ , $x_{0} = 0$ , and $ω = 1$ .

1.1F3F4F5givenalgebra

Let $1 = (1, 1, 1)^{T}$ . By [F4], $A = \frac{1}{4} (I + 11^{T})$ . Thus $A 1 = 1$ , and for every $v ⊥ 1$ one has $1^{T} v = 0$ and $A v = \frac{1}{4} v$ . These orthogonal subspaces span $R^{3}$ , so [F3]–[F5] give $∥ A ∥ = 1$ and $0 < 1 < 2 / ∥ A ∥^{2}$ .

2.1A1F1F2step 1.1algebra∎

Since $x^{†}$ is a zero-sum eigenvector, $A^{T} A x^{†} = \frac{1}{16} x^{†}$ . Applying [F2] gives $x_{k} = [1 - (1 - 1 / 16)^{k}] x^{†} = [1 - (15 / 16)^{k}] x^{†}$ . At $k = 1$ this is $\frac{1}{16} (1, 0, - 1)^{T}$ , and the scalar coefficient tends to $1$ as $k \to \infty$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

34 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Christian Clason, Regularization of Inverse Problems, full lecture notes (standard reference, not scraped)