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

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Kernel ambiguity, an unstable diagonal inverse, and a constrained class

Example

For $n = 1, 2$ , give $R^{n}$ the metric $d_{1} (x, z) = \sum_{k < n} ∣ x_{k} - z_{k} ∣$ , using the coordinatewise real vector-space structure. In the first model take parameter space and admissible truth class $X = U = R^{2}$ , data space $Y = R^{1}$ , and identity recovery target $Q = U$ with $q = {id}_{U}$ and the restricted $d_{1}$ metric. Define $A : R^{2} \to R^{1}$ by $A (a, b) = (a)$ , take $F : = A$ , and define $\ker A : = {x \in R^{2} : A (x) = (0)}$ . Then $\ker A = {(0, t) : t \in R}$ is nontrivial. The truths $x_{0} = (0, 0)$ and $x_{1} = (0, 1)$ have the same exact datum $(0)$ and target-class distance $d_{1} (x_{0}, x_{1}) = 1$ , so every deterministic data-only estimator at that datum has worst-case loss at least $1 / 2$ .

Let $N = {0, 1, 2, \dots}$ and use the real Hilbert space $ℓ^{2} (N, R)$ . For each $n \in N$ , let $e_{n}$ be the sequence with value $1$ at $n$ and $0$ elsewhere. Define the diagonal map $D : ℓ^{2} \to ℓ^{2}$ by $(D x)_{n} = (1 / 2)^{n} x_{n}$ . In the second model take $X = U = Y = Q = ℓ^{2} (N, R)$ with its canonical norm metric $d_{2} (x, z) = ∥ x - z ∥_{2}$ , forward map $F : = D$ , and identity target map $q = {id}_{U}$ . This map is well-defined, linear, and injective, hence identifiable, but it has no uniform stability modulus on all of $ℓ^{2}$ : $∥ e_{n} ∥_{2} = 1$ and $∥ D e_{n} ∥_{2} = (1 / 2)^{n} \to 0$ . For each $n$ , the observed datum $y = 0$ lies within noise radius $δ = (1 / 2)^{n}$ of both $D (0)$ and $D (e_{n})$ , so every deterministic data-only estimator at $y = 0$ has worst-case loss at least $1 / 2$ .

Finally, restrict $A$ to $U_{0} : = {(a, 0) : a \in R}$ and use $U_{0}$ itself as the target class with its inherited $d_{1}$ metric. The restricted map is identifiable, and its inverse $\hat{q} ((a)) = (a, 0)$ is an isometry; the function $ω (t) = t$ is a uniform stability modulus. This restriction removes the particular kernel ambiguity above.

Example references

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

Giannis Daras et al., A Survey on Diffusion Models for Inverse Problems, §1.2

Facts & Assumptions

Given: The metric and vector-space structures in the example statement, the canonical norm on real $ℓ^{2}$ , the geometric sequence $((1 / 2)^{n})$ , and the theorem on recovery from two truths compatible with a common datum.

[F1]

The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ : $R^{n}$ is the function space on $n$ with its coordinatewise real vector-space operations.

[F2]

Linear map between vector spaces over the same field: a map $T : V \to W$ between real vector spaces is linear when $T (a u + b v) = a T (u) + b T (v)$ for all real $a, b$ and vectors $u, v$ .

[F3]

$R^{n}$ as the set of functions $n \to R$ , and $d_{1}$ , $d_{2}$ , $d_{\infty}$ are metrics on it: for $n \geq 1$ , $d_{1} (x, z) = \sum_{k < n} ∣ x_{k} - z_{k} ∣$ is a metric on $R^{n}$ .

[F4]

Inverse-problem models and solution classes: the solution-class target is a metric space $(Q, d_{Q})$ with map $q : U \to Q$ ; the identity target is allowed.

[F5]

Identifiability and stability moduli: identifiability is equal data implying equal target classes, and a uniform stability modulus bounds target-class distance as a function of data distance.

[F6]

Kernel directions preclude unrestricted recovery: if one datum lies within radius $δ$ of both forward data, every estimator's maximum target-class loss is at least half the target-class distance.

[F7]

The real square-summable sequence space is complete: $ℓ^{2} (N, R)$ consists of sequences with finite sum of squares, uses its canonical inner-product norm, and is a real Hilbert space.

[F8]

Square-summable families on an arbitrary index set and the space $ℓ^{2} (I)$ : for a nonnegative family, its sum is the supremum of its finite subsums.

[F9]

Square-summable families on an arbitrary index set and the space $ℓ^{2} (I)$ : $e_{n}$ is the family that is $1$ at coordinate $n$ and $0$ elsewhere.

[F10]

Hilbert space: a real Hilbert space is a real inner-product space complete for its induced norm metric.

[F11]

For $∣ r ∣ < 1$ the sequence $r^{k}$ is null, and for $∣ r ∣ > 1$ the sequence $∣ r ∣^{k}$ diverges to $+ \infty$ : if $∣ r ∣ < 1$ then $r^{n} \to 0$ , and if $∣ r ∣ > 1$ then $∣ r ∣^{n}$ diverges to $+ \infty$ .

[F12]

Monotonicity of $x \mapsto x^{n}$ and of $n \mapsto a^{n}$ : for $a > 0$ all natural powers $a^{n}$ are positive, and for $0 \leq a \leq 1$ they satisfy $a^{n} \leq 1$ .

Verification

1.1F3F7F10

The $d_{1}$ formulas on $R^{1}$ and $R^{2}$ are metrics by [F3]. The real $ℓ^{2}$ space is a Hilbert space by [F7] and [F10], so its canonical norm metric is $d_{2} (x, z) : = ∥ x - z ∥_{2}$ . These are the metrics used in the three models.

1.2F1F2

The coordinate projection $A (a, b) = (a)$ is linear by coordinatewise addition and scalar multiplication, so $A (a, b) = (0)$ exactly when $a = 0$ ; hence $\ker A = {(0, t) : t \in R}$ and $(0, 1) \neq (0, 0)$ lies in its kernel.

1.3F2F7F8F12algebra

For every $x \in ℓ^{2}$ and finite $E \subseteq N$ , [F12] gives $\sum_{n \in E} ∣ (D x)_{n} ∣^{2} = \sum_{n \in E} (1 / 2)^{2 n} ∣ x_{n} ∣^{2} \leq \sum_{n \in E} ∣ x_{n} ∣^{2} \leq ∥ x ∥_{2}^{2}$ . Taking the supremum over finite $E$ by [F8] shows $\sum_{n} ∣ (D x)_{n} ∣^{2} \leq ∥ x ∥_{2}^{2} < \infty$ , hence $D x \in ℓ^{2}$ by [F7]. For real scalars $a, b$ and sequences $x, z$ , each coordinate satisfies $(D (a x + b z))_{n} = (1 / 2)^{n} (a x_{n} + b z_{n}) = a (D x)_{n} + b (D z)_{n}$ ; equality of all coordinates gives $D (a x + b z) = a D x + b D z$ , so $D$ is linear by [F2].

1.4F5F7F9F11F12algebra

Since every coefficient $(1 / 2)^{n}$ is positive by [F12], $D x = 0$ implies $x_{n} = 0$ for every $n$ , so $D$ is injective. For each $n$ , $D e_{n} = (1 / 2)^{n} e_{n}$ and hence $∥ D e_{n} ∥_{2} = (1 / 2)^{n} \to 0$ by [F11], while $∥ e_{n} ∥_{2} = 1$ by [F7, F9]. At $n = 0$ the coefficient is $1$ , so the first coordinate is unchanged; the decay is as $n \to \infty$ . If a uniform stability modulus $ω$ existed, [F5] applied to $0, e_{n}$ would give $1 \leq ω ((1 / 2)^{n})$ for all $n$ ; but the modulus limit supplies $η > 0$ with $0 < t < η \Rightarrow ω (t) < 1 / 2$ , and [F11] supplies $n$ with $0 < (1 / 2)^{n} < η$ , a contradiction.

2.1F3F4F6step 1.2algebra

We have $A (x_{0}) = A (x_{1}) = (0)$ and $d_{1} (x_{0}, x_{1}) = ∣ 0 - 0 ∣ + ∣ 0 - 1 ∣ = 1$ . Applying the two-truth theorem with $y = (0)$ and $δ = 0$ gives maximum loss at least $1 / 2$ for every estimator.

2.2F4F6F7F9step 1.4algebra

For each $n$ , $d_{2} (0, D (0)) = 0 \leq (1 / 2)^{n}$ and $d_{2} (0, D (e_{n})) = (1 / 2)^{n}$ . The target truths have distance $d_{2} (0, e_{n}) = 1$ , so the two-truth theorem at common datum $y = 0$ and radius $δ = (1 / 2)^{n}$ gives maximum estimator loss at least $1 / 2$ .

3.1F1F3F4F5step 1.2algebra∎

On $U_{0}$ , $A (a, 0) = (a)$ , so equality of data implies equality of truths. For $a, b \in R$ , $d_{1} ((a, 0), (b, 0)) = ∣ a - b ∣ = d_{1} ((a), (b)) = d_{Y} (A (a, 0), A (b, 0))$ ; therefore the inverse $\hat{q} ((a)) = (a, 0)$ is isometric, and $ω (t) = t$ satisfies the modulus conditions in [F5]. This is only a claim about the specified restricted class.

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Sources

Christian Clason, Regularization of Inverse Problems, Chapter 3 §§3.1–3.2 and Chapter 4 §4.1 (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)
Giannis Daras et al., A Survey on Diffusion Models for Inverse Problems, §1.2 (scope context) (standard reference, not scraped)