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

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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.

Depends on

Used by

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Sources

Litao Ding and Peter Mathé, Minimax rates for statistical inverse problems under general source conditions, Lemma 1 (standard reference, not scraped)
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 (standard reference, not scraped)
Iain M. Johnstone, Gaussian estimation: Sequence and wavelet models, §3.2 (standard reference, not scraped)