DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedverified 2026-10-22 (gpt-6.1-sol)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Weighted ellipsoid class for sequence inversion

Definition

Let $Q \in [0, \infty)$ and let $a = (a_{j})_{j \in N_{> 0}}$ be a nondecreasing sequence of positive real numbers, so $0 < a_{1} \leq a_{2} \leq \dots$ . The statistical ellipsoid of radius $Q$ and weights $a$ in the Gaussian sequence inverse model is $Θ_{a} (Q) = {θ \in ℓ^{2} (N_{> 0}, R) : \sum_{j = 1}^{\infty} a_{j}^{2} θ_{j}^{2} \leq Q^{2}} .$ No assumption that $a_{j} \to \infty$ is included. In particular, this definition does not assert that the ellipsoid is compact in $ℓ^{2}$ .

Definition references

Gaussian sequence inverse model

Square-summable sequence convention

Litao Ding and Peter Mathé, Minimax rates for statistical inverse problems under general source conditions, Definition 1, equation (4), PDF p. 1

Iain M. Johnstone, Gaussian estimation: Sequence and wavelet models, Chapter 3, ellipsoid definition (3.4), printed p. 58; §3.1 begins on p. 57

Depends on

Used by

Dependency tree · two levels

25 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Litao Ding and Peter Mathé, Minimax rates for statistical inverse problems under general source conditions, Definition 1, equation (4), PDF p. 1 (standard reference, not scraped)
Iain M. Johnstone, Gaussian estimation: Sequence and wavelet models, Chapter 3, ellipsoid definition (3.4), printed p. 58; §3.1 begins on p. 57 (standard reference, not scraped)