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Gaussian sequence inversion and exact risk — Examples and Counterexamples

1 · Prerequisites

2 · Logical flowchart

3 · Definitions, theorems and proofs

None yet.

4 · Examples, counterexamples and false statements

ExampleConstruction: AI-generatedVerification: AI-generatedverified 2026-10-22 (gpt-6.1-sol)Open item page →

Exact cutoff risk for a polynomial singular spectrum

Example

Assume Countable Choice for the upstream sequence-model coordinates. Let $p, r > 0$ , $Q \geq 0$ , $ϵ \geq 0$ , and $D \in N_{0}$ . Use the polynomial spectrum $s_{j} = j^{- p}$ and ellipsoid weights $a_{j} = j^{r}$ . In the operator taxonomy used here, this polynomial singular-value decay is called moderately ill-posed. The exact worst-case squared sequence risk of the cutoff at $D$ is $\frac{Q^{2}}{(D + 1)^{2 r}} + ϵ^{2} \sum_{j = 1}^{D} j^{2 p},$ with an empty sum when $D = 0$ .

Example references

Gaussian sequence inverse model

Weighted ellipsoid class

Exact cutoff-risk theorem

Litao Ding and Peter Mathé, Minimax rates for statistical inverse problems under general source conditions, Definition 3 (ill-posedness of the operator), PDF p. 6

Facts & Assumptions

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): Countable Choice is used upstream to choose the compact SVD coordinates and identify the complete-basis coefficient space.

[F1]

Exact worst-case risk of the truncated estimator: The worst-case squared sequence risk is $Q^{2} / a_{D + 1}^{2} + ϵ^{2} \sum_{j = 1}^{D} s_{j}^{- 2}$ for positive nondecreasing weights and $D \in N_{0}$ .

[F2]

Gaussian sequence inverse model: The normalized observation variance in coordinate $j$ is $ϵ^{2} s_{j}^{- 2}$ ; at zero noise the observation is exact.

[F3]

Weighted ellipsoid class for sequence inversion: The weights must be positive and nondecreasing.

Given: Countable Choice, the Gaussian sequence inverse model, weights $a_{j} = j^{r}$ , singular numbers $s_{j} = j^{- p}$ , $p, r > 0$ , noise $ϵ \geq 0$ , radius $Q \geq 0$ , and cutoff $D \in N_{0}$ .

Verification

1.1givenA1F2F3algebra

By [A1], the upstream sequence coordinates are fixed. Since $p > 0$ , $s_{j} = j^{- p}$ is positive and decreases to $0$ , and $r > 0$ makes $a_{j} = j^{r}$ positive and nondecreasing; thus the exact-risk theorem's spectrum and ellipsoid hypotheses hold.

2.1F1step 1.1algebra∎

Substituting $a_{D + 1} = (D + 1)^{r}$ and $s_{j}^{- 2} = j^{2 p}$ into [F1] gives the displayed risk, including $D = 0$ by its empty-sum convention.

ExampleConstruction: AI-generatedVerification: AI-generatedverified 2026-10-22 (gpt-6.1-sol)Open item page →

Exact cutoff risk for an exponentially decaying spectrum

Example

Assume Countable Choice for the upstream sequence-model coordinates. Let $c, r > 0$ , $Q \geq 0$ , $ϵ \geq 0$ , and $D \in N_{0}$ . Use the exponential spectrum $s_{j} = e^{- c j}$ and ellipsoid weights $a_{j} = j^{r}$ . This is a severely ill-posed operator spectrum, with exponential singular-value decay. The exact worst-case squared sequence risk of the cutoff at $D$ is $\frac{Q^{2}}{(D + 1)^{2 r}} + ϵ^{2} \sum_{j = 1}^{D} e^{2 c j},$ with an empty sum when $D = 0$ .

Example references

Gaussian sequence inverse model

Weighted ellipsoid class

Exact cutoff-risk theorem

Litao Ding and Peter Mathé, Minimax rates for statistical inverse problems under general source conditions, Definition 3 (ill-posedness of the operator), PDF p. 6

Facts & Assumptions

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): Countable Choice is used upstream to choose the compact SVD coordinates and identify the complete-basis coefficient space.

[F1]

[F2]

Gaussian sequence inverse model: The normalized observation variance in coordinate $j$ is $ϵ^{2} s_{j}^{- 2}$ ; at zero noise the observation is exact.

[F3]

Weighted ellipsoid class for sequence inversion: The weights must be positive and nondecreasing.

Given: Countable Choice, the Gaussian sequence inverse model, weights $a_{j} = j^{r}$ , singular numbers $s_{j} = e^{- c j}$ , with $c, r > 0$ , noise $ϵ \geq 0$ , radius $Q \geq 0$ , and cutoff $D \in N_{0}$ .

Verification

1.1givenA1F2F3algebra

By [A1], the upstream sequence coordinates are fixed. Since $c > 0$ , $s_{j} = e^{- c j}$ is positive and decreases to $0$ , and $r > 0$ makes $a_{j} = j^{r}$ positive and nondecreasing; thus the exact-risk theorem's spectrum and ellipsoid hypotheses hold.

2.1F1step 1.1algebra∎

Substituting $a_{D + 1} = (D + 1)^{r}$ and $s_{j}^{- 2} = e^{2 c j}$ into [F1] gives the displayed risk, including $D = 0$ by its empty-sum convention.

ExampleConstruction: AI-generatedVerification: AI-generatedverified 2026-10-22 (gpt-6.1-sol)Open item page →

The zero-noise boundary of truncated risk

Example

Assume Countable Choice for the upstream sequence-model coordinates. Take $ϵ = 0$ and $Q > 0$ . In the zero-noise sequence model, $Z_{j} = θ_{j}$ exactly, so the cutoff error is the deterministic squared tail $\sum_{j > D} θ_{j}^{2}$ . For weights $a_{j} = j^{r}$ with $r > 0$ , its worst case over $Θ_{a} (Q)$ is $Q^{2} / (D + 1)^{2 r}$ and tends to $0$ as $D \to \infty$ . For the equally weighted ellipsoid $a_{j} = 1$ , the worst case is $Q^{2}$ for every finite $D$ , attained by $θ = Q e_{D + 1}$ . This is a deterministic boundary of the sequence-risk formula; it makes no claim about arbitrary bounded perturbations of the original inverse data.

Example references

Gaussian sequence inverse model and its zero-noise extension

Weighted ellipsoid class

Exact cutoff-risk theorem

Facts & Assumptions

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): Countable Choice is used upstream to choose the compact SVD coordinates and identify the complete-basis coefficient space.

[F1]

Gaussian sequence inverse model: At $ϵ = 0$ , $Z_{j} = θ_{j}$ deterministically.

[F2]

Weighted ellipsoid class for sequence inversion: For any positive nondecreasing weights, $Θ_{a} (Q)$ is defined by $\sum_{j} a_{j}^{2} θ_{j}^{2} \leq Q^{2}$ .

[F3]

Exact worst-case risk of the truncated estimator: The zero-noise worst-case squared error is $Q^{2} / a_{D + 1}^{2}$ , attained by a truth supported only at coordinate $D + 1$ .

Given: Countable Choice, $Q > 0$ , a finite cutoff $D \in N_{0}$ , and either weights $a_{j} = j^{r}$ with $r > 0$ or $a_{j} = 1$ for all $j$ .

Verification

1.1A1F1F2F3algebra

Under [A1], the sequence model coordinates are fixed. Using [F1] and substituting $ϵ = 0$ and $a_{D + 1} = (D + 1)^{r}$ in [F3] gives $Q^{2} / (D + 1)^{2 r}$ ; since $r > 0$ , this tends to $0$ as $D \to \infty$ .

2.1A1F2F3algebraconstruct∎

Under [A1], [F3] applies to the unit-weight ellipsoid. The ellipsoid condition in [F2] is $∥ θ ∥_{2}^{2} \leq Q^{2}$ , so the squared tail is at most $Q^{2}$ . The vector $θ = Q e_{D + 1}$ belongs to the ellipsoid and its entire squared norm lies beyond $D$ , so it attains $Q^{2}$ . This is also the $ϵ = 0$ , $a_{D + 1} = 1$ case of [F3].

Gaussian sequence inversion and exact risk — Examples and Counterexamples

2 · Logical flowchart

3 · Definitions, theorems and proofs

4 · Examples, counterexamples and false statements

Exact cutoff risk for a polynomial singular spectrum

Example

Example references

Facts & Assumptions

Verification

Exact cutoff risk for an exponentially decaying spectrum

Example

Example references

Facts & Assumptions

Verification

The zero-noise boundary of truncated risk

Example

Example references

Facts & Assumptions

Verification

Sources