DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)

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.

Fixed-design regression with sub-Gaussian noise

Definition

Let $n, d \geq 1$ be integers, let $X \in R^{n \times d}$ be deterministic, and let $θ^{*} \in R^{d}$ . The fixed-design regression model is $Y = X θ^{*} + ε,$ where $Y, ε$ are random vectors in $R^{n}$ . For $σ^{2} \geq 0$ , the noise is centered sub-Gaussian with proxy $σ^{2}$ when $E e^{λ ⟨ u, ε ⟩} \leq e^{σ^{2} λ^{2} ∥ u ∥_{2}^{2} / 2} (u \in R^{n}, λ \in R),$ using the vector convention of Sub-Gaussian variables and vectors, and sub-exponential variables. No independence among the coordinates of $ε$ is assumed. The parameter is $s$ -sparse when $1 \leq s \leq d$ and $∣ {j : θ_{j}^{*} \neq 0} ∣ \leq s$ .

Depends on

Sub-Gaussian variables and vectors, and sub-exponential variables

Used by

Dependency tree · two levels

7 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

Philippe Rigollet and Jan-Christian Hütter, High-Dimensional Statistics: Lecture Notes, Chapter 2, Section 2.4 (standard reference, not scraped)
Peter J. Bickel, Ya'acov Ritov and Alexandre B. Tsybakov, Simultaneous Analysis of Lasso and Dantzig Selector, Section 7 (standard reference, not scraped)