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

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Analytic ellipsoid rates in polynomial and exponential spectra

Example

Assume Countable Choice, take $Q > 0$ and analytic-type weights $a_{j} ≍ e^{j}$ in the Gaussian sequence inverse model. If the singular values satisfy $s_{j} ≍ j^{- 1}$ , the minimax RMS error is of order $ϵ [\log (1 / ϵ)]^{3 / 2}$ as $ϵ ↓ 0$ ; if instead $s_{j} ≍ e^{- j}$ , it is of order $ϵ^{1 / 2}$ . The comparison constants may depend on the fixed ellipsoid and axis constants.

Example references

Polynomial and exponential minimax estimation rates

Litao Ding and Peter Mathé, Minimax rates for statistical inverse problems under general source conditions, §3 and Table 1

Facts & Assumptions

[F1]

Polynomial and exponential minimax estimation rates: For polynomial singular values and exponential ellipsoid weights, $R_{*} ≍ ϵ [\log (1 / ϵ)]^{p + 1 / 2}$ ; for exponential singular values and exponential weights, $R_{*} ≍ ϵ^{κ / (p + κ)}$ .

[F2]

Gaussian sequence inverse model and Weighted ellipsoid class for sequence inversion: The model uses the positive singular sequence and the parameter class $\sum_{j} a_{j}^{2} θ_{j}^{2} \leq Q^{2}$ .

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): Countable Choice is inherited from the sequence model.

Given: Countable Choice, $Q > 0$ , $p = κ = 1$ , and either the polynomial or exponential singular-value regime in the Example.

Verification

1.1givenF1algebra

In the polynomial regime, $p = κ = 1$ , so the first formula of [F1] is $ϵ [\log (1 / ϵ)]^{1 + 1 / 2} = ϵ [\log (1 / ϵ)]^{3 / 2}$ .

1.2givenF1algebra

In the exponential regime, the second formula of [F1] has exponent $κ / (p + κ) = 1 / 2$ , giving $ϵ^{1 / 2}$ .

2.1A1step 1.1step 1.2F2algebra∎

By [A1], the Gaussian sequence model is interpreted with its stated Countable Choice prerequisite. The two substitutions give exactly the two stated RMS orders; $Q$ and the asymptotic comparison constants affect only the multiplicative bounds.

Depends on

Used by

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Sources

Litao Ding and Peter Mathé, Minimax rates for statistical inverse problems under general source conditions, §3 and Table 1, PDF pp. 5–6 (standard reference, not scraped)