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Inverse maps, identifiability, and stability

1 · Prerequisites

2 · Logical flowchart

3 · Definitions, theorems and proofs

DefinitionDefinition: AI-adaptedProof: Not applicablejudge pass (gpt-6-luna)verified 2026-10-22 (gpt-6.1-sol)Open item page →

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

DefinitionDefinition: AI-adaptedProof: Not applicablejudge pass (gpt-6-luna)verified 2026-10-22 (gpt-6.1-sol)Open item page →

Identifiability and stability moduli

Definition

Use metric spaces $(X, d_{X})$ and $(Y, d_{Y})$ , a nonempty admissible truth class $U \subseteq X$ , a forward map $F : U \to Y$ , and a metric solution-class space $(Q, d_{Q})$ with class map $q : U \to Q$ as follows: $q$ has image all of $Q$ , and $q (x) = q (x^{'})$ exactly when $x$ and $x^{'}$ are equivalent for recovery. The library's metric axioms imply nonnegative metric values.

The forward map $F$ is identifiable modulo the recovery equivalence on $U$ if

$x, x^{'} \in U and F (x) = F (x^{'}) ⟹ q (x) = q (x^{'}) .$

It is nonidentifiable modulo that equivalence if some two admissible truths have equal data and distinct target classes.

A deterministic data-only estimator is a function $\hat{q} : Y \to Q$ . It is exact on $U$ if $\hat{q} (F (x)) = q (x)$ for every $x \in U$ . The diagram below commutes exactly under this condition; it states the defining equality of functions and is not a proof device.

A uniform stability modulus on $U$ is a nondecreasing function $ω : [0, \infty) \to [0, \infty)$ , with $ω (0) = 0$ and $\lim_{t ↓ 0} ω (t) = 0$ , such that

$d_{Q} (q (x), q (x^{'})) \leq ω (d_{Y} (F (x), F (x^{'}))) (x, x^{'} \in U) .$

Identifiability concerns equality of exact data; the modulus is a separate quantitative bound for all truth pairs. The estimator definition specifies no regularization procedure or convergence guarantee.

Definition references

Inverse-problem model and solution-class interface

Metric-space axioms

Nonnegative values of metric distances

Christian Clason, Regularization of Inverse Problems, Chapter 3 §§3.1–3.2 and Chapter 4 §4.1

Tristan van Leeuwen and Christoph Brune, 10 Lectures on Inverse Problems and Imaging, Lecture 1 §§1.2–1.3

TheoremStatement: AI-adaptedProof: AI-adaptedjudge pass (gpt-6-luna)verified 2026-10-22 (gpt-6.1-sol)Open item page →

Kernel directions preclude unrestricted recovery

Statement

Let $(X, d_{X})$ , $(Y, d_{Y})$ , and $(Q, d_{Q})$ be metric spaces, let $U \subseteq X$ be a nonempty admissible truth class, let $F : U \to Y$ be a forward map, and let $q : U \to Q$ assign each truth its target class. Let $x_{0}, x_{1} \in U$ , let $δ \geq 0$ , and suppose a single datum $y \in Y$ satisfies

$d_{Y} (y, F (x_{0})) \leq δ and d_{Y} (y, F (x_{1})) \leq δ .$

For every deterministic data-only estimator, meaning every function $\hat{q} : Y \to Q$ , the two target-class losses at that datum satisfy

$\max {d_{Q} (\hat{q} (y), q (x_{0})), d_{Q} (\hat{q} (y), q (x_{1}))} \geq \frac{1}{2} d_{Q} (q (x_{0}), q (x_{1})) .$

In particular, if $F (x_{0}) = F (x_{1})$ but $q (x_{0}) \neq q (x_{1})$ , no data-only estimator exactly recovers both truths. The quantitative bound also holds whenever the radius- $δ$ data balls around the two forward data share the observed datum $y$ .

Statement references

Inverse-problem model and target-class loss

Identifiability and deterministic estimators

Metric triangle inequality

Nonnegative metric distances

Christian Clason, Regularization of Inverse Problems, Chapter 3 and Chapter 4 §4.1

Tristan van Leeuwen and Christoph Brune, 10 Lectures on Inverse Problems and Imaging, Lecture 1 §§1.2–1.3

Facts & Assumptions

Given: Metric spaces $(X, d_{X})$ , $(Y, d_{Y})$ , and $(Q, d_{Q})$ ; a nonempty set $U \subseteq X$ ; a map $F : U \to Y$ ; a target-class map $q : U \to Q$ ; truths $x_{0}, x_{1} \in U$ ; a radius $δ \geq 0$ ; a datum $y \in Y$ within distance $δ$ of both $F (x_{0})$ and $F (x_{1})$ ; and an estimator $\hat{q} : Y \to Q$ .

[F1]

Inverse-problem models and solution classes: the recovery target has a metric $d_{Q}$ , and $q (x)$ is the target class of truth $x$ .

[F2]

Identifiability and stability moduli: a deterministic data-only estimator is a function $Y \to Q$ and exact recovery requires $\hat{q} (F (x)) = q (x)$ for every admissible truth.

[F3]

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

[F4]

Nonnegativity of a metric is a consequence of the other axioms, not an axiom: every metric distance is nonnegative.

Proof

1.1givenF2

Put $z : = \hat{q} (y) \in Q$ . Since the estimator is a function evaluated at the single datum $y$ , this is the same estimate for both truths.

1.2F1F3

By the triangle inequality for $d_{Q}$ , $d_{Q} (q (x_{0}), q (x_{1})) \leq d_{Q} (q (x_{0}), z) + d_{Q} (z, q (x_{1})) .$

2.1step 1.2F4algebra

Let $M$ be the maximum of the two losses in the statement. Both losses are nonnegative by [F4] and each summand in step 1.2 is at most $M$ , so $d_{Q} (q (x_{0}), q (x_{1})) \leq 2 M$ and $M \geq \frac{1}{2} d_{Q} (q (x_{0}), q (x_{1}))$ .

3.1givenF2F3F4step 1.1step 2.1∎

If $F (x_{0}) = F (x_{1})$ and their target classes differ, metric separation and nonnegativity give $d_{Q} (q (x_{0}), q (x_{1})) > 0$ ; choosing $y = F (x_{0})$ and $δ = 0$ satisfies the hypotheses, so step 2.1 rules out exact recovery of both. For a shared noisy datum the same estimator output from step 1.1 yields the same bound.

4 · Examples, counterexamples and false statements

None yet.

Inverse maps, identifiability, and stability

2 · Logical flowchart

3 · Definitions, theorems and proofs

Inverse-problem models and solution classes

Definition

Definition references

Identifiability and stability moduli

Definition

Definition references

Kernel directions preclude unrestricted recovery

Statement

Statement references

Facts & Assumptions

Proof

4 · Examples, counterexamples and false statements

Sources