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 incoherence condition

Definition

For integers $n, d, s$ with $n \geq 1$ and $1 \leq s \leq d$ , and a deterministic matrix $X \in R^{n \times d}$ , write $G = X^{T} X / n$ . The design satisfies the incoherence condition at sparsity $s$ if $\max_{1 \leq j, k \leq d} ∣ G_{j k} - 1_{{j = k}} ∣ \leq \frac{1}{14 s} .$ Here the maximum is entrywise, $1_{{j = k}}$ is $1$ when $j = k$ and $0$ otherwise, and no probabilistic assumption on $X$ is made. The constant $1 / (14 s)$ is the exact premise used in the local restricted-eigenvalue derivation; it is not a generic definition of design compatibility.

Used by

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

Philippe Rigollet, 18.S997 High-Dimensional Statistics, Chapter 2, Section 2.4 (standard reference, not scraped)
Philippe Rigollet and Jan-Christian Hütter, High-Dimensional Statistics: Lecture Notes, Chapter 2, Section 2.4 (standard reference, not scraped)