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.

Restricted eigenvalue condition on the Lasso cone

Definition

Let $n, d \geq 1$ , $1 \leq s \leq d$ , and $X \in R^{n \times d}$ . For $S \subseteq {1, \dots, d}$ , let $v_{S}$ be the vector agreeing with $v \in R^{d}$ on $S$ and equal to zero off $S$ ; write $∥ v ∥_{1} = \sum_{j} ∣ v_{j} ∣$ and use the Euclidean norm $∥ \cdot ∥_{2}$ of Euclidean spheres and closed balls as subspaces of $R^{n}$ . The cone associated to $S$ is $C (S) = {v \in R^{d} : ∥ v_{S^{c}} ∥_{1} \leq 3 ∥ v_{S} ∥_{1}} .$ The design satisfies the restricted eigenvalue condition with constant $κ^{2} > 0$ at sparsity $s$ when $\frac{∥ X v ∥_{2}^{2}}{n} \geq κ^{2} ∥ v_{S} ∥_{2}^{2}$ for every $S$ with $∣ S ∣ \leq s$ and every nonzero $v \in C (S)$ . The denominator is the norm of the supported component; this condition does not assert a lower bound for all vectors in $R^{d}$ .

Depends on

Euclidean spheres and closed balls as subspaces of $\mathbb{R}^n$

Used by

Dependency tree · two levels

5 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 3 (standard reference, not scraped)