DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)verified 2026-10-22 (gpt-6.1-sol)

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.

Noise-level-free parameter-choice rule

Definition

Fix a compact Hilbert inverse model $K : X \to Y$ and a family of reconstruction maps $(R_{α})_{α > 0}$ , with each $R_{α} : Y \to X$ , both held fixed. A deterministic parameter-choice rule is a function $α : Y \to (0, \infty)$ . It is noise-level-free (also called heuristic) when the value $α (z)$ for an observed datum $z \in Y$ uses only $K$ , the fixed family, and $z$ ; it does not take the exact datum or a noise bound $δ$ as input. Here $δ \geq 0$ is a valid noise bound for $z$ and an exact datum $y$ when $∥ z - y ∥_{Y} \leq δ$ . The resulting reconstruction is $R_{α (z)} z$ .

Definition references

Compact Hilbert inverse model

Clason, Regularization of Inverse Problems, Definition 4.4

Sherry, Inverse Problems: Lecture Notes 2024, Section 3.2.3

Depends on

Compact linear inverse model between Hilbert spaces

Used by

Noise underestimation defeats a truncated singular-value rule Counterexample
Bakushinskii veto for discontinuous pseudoinverses Theorem

Dependency tree · two levels

4 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: Lecture Notes, Definition 4.4 (standard reference, not scraped)
Ferdia Sherry, Inverse Problems: Lecture Notes 2024, Section 3.2.3 (standard reference, not scraped)