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Minimax estimation over inverse source classes — Examples

1 · Prerequisites

2 · Logical flowchart

3 · Definitions, theorems and proofs

None yet.

4 · Examples, counterexamples and false statements

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

Sobolev ellipsoid rates in polynomial and exponential spectra

Example

Assume Countable Choice, take $Q > 0$ and Sobolev-type weights $a_{j} ≍ j$ in the Gaussian sequence inverse model. If the singular values satisfy $s_{j} ≍ j^{- 1}$ , the minimax RMS error is of order $ϵ^{2 / 5}$ as $ϵ ↓ 0$ ; if instead $s_{j} ≍ e^{- j}$ , it is of order $[\log (1 / ϵ)]^{- 1}$ . 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 polynomial ellipsoid weights, $R_{*} ≍ ϵ^{κ / (κ + p + 1 / 2)}$ ; for exponential singular values and polynomial weights, $R_{*} ≍ [\log (1 / ϵ)]^{- κ}$ .

[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, $p = 1$ and $κ = 1$ , so the first formula of [F1] has exponent $1 / (1 + 1 + 1 / 2) = 2 / 5$ .

1.2givenF1algebra

In the exponential regime, $κ = 1$ , so the second formula of [F1] is $[\log (1 / ϵ)]^{- 1}$ .

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.

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

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∎

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

First-order power-source rates in polynomial and exponential spectra

Example

Assume Countable Choice. Let $X$ and $Y$ be real Hilbert spaces, let $K : X \to Y$ be a compact operator of infinite rank, and write $s_{1} \geq s_{2} \geq \dots > 0$ , $s_{j} \to 0$ , for its positive singular values. Set $Q = ρ > 0$ and $ν = 1$ . Over the minimum-norm power source ball $X_{1, ρ} (K)$ in the Gaussian sequence inverse model associated with this singular system, if $s_{j} ≍ j^{- 1}$ then the minimax RMS error is of order $ϵ^{2 / 5}$ ; if $s_{j} ≍ e^{- j}$ then it is of order $ϵ^{1 / 2}$ as $ϵ ↓ 0$ . The constants may depend on $ρ$ and the fixed singular-value comparison constants.

Example references

Power source balls are weighted statistical ellipsoids

Polynomial and exponential minimax estimation rates

Litao Ding and Peter Mathé, Minimax rates for statistical inverse problems under general source conditions, Definition 2 and §3, Table 1

Facts & Assumptions

[F1]

Power source balls are weighted statistical ellipsoids: $X_{ν, ρ} (K)$ has ellipsoid weights $a_{j} = s_{j}^{- ν}$ and radius $Q = ρ$ .

[F2]

Polynomial and exponential minimax estimation rates: For polynomial singular values with polynomial axes, the rate is $ϵ^{κ / (κ + p + 1 / 2)}$ ; for exponential singular values with exponential axes, the rate is $ϵ^{κ / (p + κ)}$ .

[F3]

Power source ball and Gaussian sequence inverse model: The source class is $X_{ν, ρ} (K) = {∣ K ∣^{ν} w : ∥ w ∥_{X} \leq ρ}$ in the real Gaussian sequence inverse model.

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): Countable Choice is inherited through the compact singular system.

Given: Countable Choice, real Hilbert spaces $X, Y$ , a compact infinite-rank operator $K : X \to Y$ with positive singular values $s_{1} \geq s_{2} \geq \dots > 0$ and $s_{j} \to 0$ , $ρ > 0$ , $ν = 1$ , and either $s_{j} ≍ j^{- 1}$ or $s_{j} ≍ e^{- j}$ .

Verification

1.1givenF1F2algebra

If $s_{j} ≍ j^{- 1}$ , [F1] gives $a_{j} = s_{j}^{- 1} ≍ j$ , so the first formula of [F2] applies with $p = κ = 1$ and gives $ϵ^{2 / 5}$ .

1.2givenF1F2algebra

If $s_{j} ≍ e^{- j}$ , [F1] gives $a_{j} = s_{j}^{- 1} ≍ e^{j}$ , so the second formula of [F2] applies with $p = κ = 1$ and gives $ϵ^{1 / 2}$ .

2.1givenA1F1F2F3step 1.1step 1.2algebra∎

By [A1], the compact singular-system coordinate model is interpreted with its stated Countable Choice prerequisite. Both substitutions use the exact coefficient ellipsoid from [F1] and preserve the stated radius $ρ$ , so they yield the two source-ball rates in the Example.

Minimax estimation over inverse source classes — Examples

2 · Logical flowchart

3 · Definitions, theorems and proofs

4 · Examples, counterexamples and false statements

Sobolev ellipsoid rates in polynomial and exponential spectra

Example

Example references

Facts & Assumptions

Verification

Analytic ellipsoid rates in polynomial and exponential spectra

Example

Example references

Facts & Assumptions

Verification

First-order power-source rates in polynomial and exponential spectra

Example

Example references

Facts & Assumptions

Verification

Sources