CounterexampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck passverified 2026-10-22 (gpt-6.1-sol)

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Noise underestimation defeats a truncated singular-value rule

Statement refuted

False claim. Let $X = Y = ℓ^{2} (N, R)$ with standard basis $(e_{j})_{j \geq 1}$ , let $K x = (x_{j} / j)_{j \geq 1}$ , and for $α > 0$ define $R_{α} z = \sum_{k = 1}^{⌊ 1 / α ⌋} k z_{k} e_{k}$ , with the empty sum used when the upper limit is $0$ . Define the data-only noise proxy by $\hat{δ} (0) = 1$ and $\hat{δ} (z) = \min {1, ∥ z ∥_{2}^{2}}$ for $z \neq 0$ , and set $α (z) = \hat{δ} (z)$ . The false claim is that this compact operator, pointwise regularizing family, and deterministic noise-level-free rule satisfy, for every exact datum $y \in D (K^{†})$ , $\lim_{δ ↓ 0} \sup {∥ R_{α (z)} z - K^{†} y ∥_{2} : z \in ℓ^{2} (N, R), ∥ z - y ∥_{2} \leq δ} = 0$ .

Statement references

Compact Hilbert inverse model

Hilbert-space Moore–Penrose inverse and its domain

Noise-level-free parameter-choice rule

Clason, Regularization of Inverse Problems, Theorem 4.8

Sherry, Inverse Problems: Lecture Notes 2024, Theorem 3.12

Facts & Assumptions

Given: $N = {1, 2, \dots}$ , the real sequence space $ℓ^{2} (N, R)$ , $K x = (x_{j} / j)_{j \geq 1}$ , its coordinate truncations $K_{N} x = (x_{1}, x_{2} / 2, \dots, x_{N} / N, 0, \dots)$ , the family $R_{α} z = \sum_{k = 1}^{⌊ 1 / α ⌋} k z_{k} e_{k}$ , and the rule $α (0) = 1$ , $α (z) = \min {1, ∥ z ∥_{2}^{2}}$ for $z \neq 0$ .

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): For every family $(X_{n})_{n \in N}$ of nonempty sets indexed by $N$ , there is a function $f$ with domain $N$ such that $f (n) \in X_{n}$ for every $n$ .

[F1]

Square-summable families on an arbitrary index set and the space $ℓ^{2} (I)$ : For $x \in ℓ^{2} (N, R)$ , $∥ x ∥_{2}^{2} = \sum_{j \geq 1} ∣ x_{j} ∣^{2}$ .

[F14]

Square-summable families on an arbitrary index set and the space $ℓ^{2} (I)$ : The coordinate vector $e_{j}$ has value $1$ in coordinate $j$ and $0$ elsewhere.

[F2]

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

[F3]

A bounded linear operator between normed spaces: A linear map $T$ between normed spaces is bounded when some real $C \geq 0$ satisfies $∥ T x ∥ \leq C ∥ x ∥$ for every $x$ .

[F4]

The operator norm as the least bound and as the unit-sphere or unit-ball supremum: For a bounded linear map $T$ , $∥ T ∥ = \sup {∥ T x ∥ : ∥ x ∥ \leq 1}$ .

[F5]

Basis of a vector space: a linearly independent spanning subset; and ordered basis: an injective finite list whose image is a basis: An ordered basis of a vector space is an injective finite list whose image is a basis.

[F6]

Bounded finite rank operators are compact: A bounded linear operator whose range has a finite ordered basis is compact.

[F7]

Banach space: A normed space is Banach when it is complete in its norm metric.

[F8]

Norm limit of compact operators is compact: Assuming Countable Choice, a norm limit of compact operators into a Banach space is compact.

[F9]

Compact linear operator: A linear map is compact when every bounded input set has image with compact closure.

[F10]

Compact linear inverse model between Hilbert spaces: The compact inverse model has real or complex Hilbert parameter and data spaces over one field and a compact linear forward map.

[F11]

Hilbert-space Moore–Penrose inverse: $D (K^{†}) = R (K) \oplus R (K)^{⊥}$ , and on $r + z$ with $r \in R (K)$ and $z \in R (K)^{⊥}$ , $K^{†} (r + z)$ is the unique minimum-norm preimage of $r$ .

[F12]

Noise-level-free parameter-choice rule: A noise-level-free parameter is a function of the observed datum and fixed operator/family only; it does not take an exact datum or noise bound as input.

[F13]

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

Counterexample

1.1F1F3F4F5F13F14

Coordinatewise linearity gives $K$ and every $K_{N}$ as linear maps by [F13]. For every $x \in ℓ^{2}$ , [F1] gives $∥ K x ∥_{2}^{2} = \sum_{j \geq 1} j^{- 2} ∣ x_{j} ∣^{2} \leq ∥ x ∥_{2}^{2}$ and $∥ K_{N} x ∥_{2}^{2} \leq ∥ x ∥_{2}^{2}$ , so [F3] makes both bounded linear operators. For $∥ x ∥_{2} \leq 1$ , [F1] also gives $∥ (K - K_{N}) x ∥_{2}^{2} = \sum_{j > N} j^{- 2} ∣ x_{j} ∣^{2} \leq (N + 1)^{- 2}$ . Hence [F4] gives $∥ K - K_{N} ∥ \leq 1 / (N + 1)$ . The range of $K_{N}$ is exactly the span of $e_{1}, \dots, e_{N}$ : its outputs are supported in these coordinates, and $K_{N} (j e_{j}) = e_{j}$ for each $j \leq N$ by [F14]. These vectors form an ordered basis by [F5], so each $K_{N}$ has finite rank.

2.1A1step 1.1F2F6F7F8F9F10

By [F6], every $K_{N}$ is compact in the sense of [F9]. By [F2] and [F7], the codomain $ℓ^{2}$ is Banach; [A1] supplies the Countable Choice hypothesis of [F8]. Since $∥ K - K_{N} ∥ \to 0$ by step 1.1, [F8] implies that $K$ is compact and hence satisfies the compact-model interface [F10].

3.1step 2.1F1F11F14algebra

The coordinate formula makes $K$ injective. If $y = K x$ , then $x_{j} = j y_{j}$ and $(j y_{j})_{j \geq 1} \in ℓ^{2}$ ; conversely, whenever $(j y_{j})_{j \geq 1} \in ℓ^{2}$ , taking $x_{j} = j y_{j}$ gives $K x = y$ . Thus $R (K) = {y : (j y_{j})_{j \geq 1} \in ℓ^{2}}$ . Each $e_{j} / j = K e_{j}$ lies in $R (K)$ , and these vectors are linearly independent because they have distinct nonzero coordinates by [F14]; hence $R (K)$ is infinite-dimensional. Any vector in $R (K)^{⊥}$ has every coordinate zero, so $R (K)^{⊥} = {0}$ and $D (K^{†}) = R (K)$ . By [F11], $K^{†} y = (j y_{j})_{j \geq 1}$ . For each fixed $α > 0$ , $R_{α}$ is bounded since its sum has finitely many terms and $∥ R_{α} y ∥_{2}^{2} \leq ⌊ 1 / α ⌋^{2} ∥ y ∥_{2}^{2}$ . For $y \in D (K^{†})$ , $R_{α} y$ is the initial truncation of the square-summable sequence $K^{†} y$ , so as $α ↓ 0$ these truncations converge in $ℓ^{2}$ to $K^{†} y$ . The stated family is therefore pointwise regularizing.

4.1givenF1F11F12F14step 3.1constructalgebra∎

The rule $α$ is positive and depends only on $z$ , so it is noise-level-free by [F12]. Take exact datum $y = 0$ and noisy datum $z_{j} = e_{j} / j = K e_{j}$ . By [F1] and [F14], $δ_{j} = ∥ z_{j} - y ∥_{2} = 1 / j \to 0$ , while $α (z_{j}) = 1 / j^{2}$ and $⌊ 1 / α (z_{j}) ⌋ = j^{2}$ . Therefore $R_{α (z_{j})} z_{j} = e_{j}$ , whereas $K^{†} y = 0$ , and the reconstruction error is $1$ for every $j$ . The claimed worst-case limit fails at $y = 0$ , since its supremum is at least $1$ along the null sequence $δ_{j}$ .

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Sources

Christian Clason, Regularization of Inverse Problems: Lecture Notes, Theorem 4.8 (standard reference, not scraped)
Ferdia Sherry, Inverse Problems: Lecture Notes 2024, Theorem 3.12 (standard reference, not scraped)