DefinitionDefinition: AI-adaptedProof: 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.

Inverse-problem models and solution classes

Definition

Let $(X, d_{X})$ and $(Y, d_{Y})$ be metric spaces, called the parameter space and the data space. Let $U \subseteq X$ be a nonempty set of admissible truths, and let $F : U \to Y$ be a forward map. For $x \in U$ , the datum $F (x)$ is its exact observation. For $y \in Y$ , define the exact solution set

$S_{F} (y) : = {x \in U : F (x) = y} .$

The datum $y$ is exactly solvable on $U$ when $S_{F} (y) \neq \emptyset$ .

Let $\sim$ be an equivalence relation on $U$ , meaning it is reflexive, symmetric, and transitive; truths equivalent under $\sim$ represent the same recovery target. A metric solution-class space for $(U, \sim)$ consists of a metric space $(Q, d_{Q})$ and a function $q : U \to Q$ whose image is all of $Q$ and such that $q (x) = q (x^{'})$ if and only if $x \sim x^{'}$ . The loss between two target classes is their metric distance $d_{Q}$ . Exact recovery of the truth itself is the special case $Q = U$ , $q (x) = x$ , and $d_{Q} = d_{X}$ restricted to $U$ .

Definition references

Metric-space axioms

Christian Clason, Regularization of Inverse Problems, Chapter 3

Tristan van Leeuwen and Christoph Brune, 10 Lectures on Inverse Problems and Imaging, Lecture 1

Depends on

Metric space: $d(x,y) = 0$ iff $x = y$, symmetry, and the triangle inequality; pseudometric and ultrametric

Used by

A Borel prior concentrated on a forward-map domain Definition
Admissible constraint classes and restricted identifiability Definition
Compact linear inverse model between Hilbert spaces Definition
Identifiability and stability moduli Definition
Inverse reconstruction risk under train and test laws Definition
Learned inverse estimators and reconstruction targets Definition
Nonlinear forward maps between Hilbert spaces Definition
Operator, noise-law, and prior-support mismatch Definition
Real phase retrieval via finite real measurements Definition
The nonlinear Tikhonov functional Definition
Weakly closed nonlinear forward maps on Hilbert spaces Definition
A Gaussian coefficient law with mass outside the uniform-ellipticity domain Example
A Gaussian wavelet series at the Besov threshold Example
A posterior mean can lie outside a nonconvex signal class Example
A proper Gaussian likelihood gives a proper posterior Example
A trace-class Gaussian with full support and a proper Cameron–Martin space Example
A train/test prior shift changes reconstruction risk Example
A zero-evidence fiber has no normalized fiberwise update Example
An exact posterior sample need not minimize point risk Example
An improper counting prior can have finite evidence at one datum Example
Kernel ambiguity, an unstable diagonal inverse, and a constrained class Example
Equal observation laws force a metric-risk lower bound Theorem
Kernel directions preclude unrestricted recovery Theorem

Dependency tree · two levels

7 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, Chapter 3, printed pp. 15–27 (standard reference, not scraped)
Tristan van Leeuwen and Christoph Brune, 10 Lectures on Inverse Problems and Imaging, Lecture 1 §§1.2–1.3 (standard reference, not scraped)