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

The ℓ2 robust nullspace property

Definition

Let $m, n, s \in N$ with $m, n \geq 1$ and $1 \leq s \leq n$ , let $A \in R^{m \times n}$ , and let $0 < ρ < 1$ and $τ > 0$ . In real scalar expressions, a natural number denotes its canonical real image. Use one-based coordinate labels $[n] = {1, \dots, n}$ , identifying coordinate label $j$ with library coordinate $j - 1$ . For $h \in R^{n}$ and $S \subseteq [n]$ , let $h_{S}$ agree with $h$ on $S$ and vanish on $S^{c} = [n] ∖ S$ . The matrix $A$ satisfies the $ℓ^{2}$ robust nullspace property of order $s$ with constants $(ρ, τ)$ if $∥ h_{S} ∥_{2} \leq \frac{ρ}{\sqrt{s}} ∥ h_{S^{c}} ∥_{1} + τ ∥ A h ∥_{2}$ for every $h \in R^{n}$ and every $S \subseteq [n]$ with $∣ S ∣ \leq s$ . This condition applies to all vectors, including vectors with $A h \neq 0$ .

Definition references

One-based coordinates and zero-extended restrictions

Finite-dimensional one- and two-norms

Canonical real image of the natural scalar $s$

Positivity of the natural scalar $s$

Mendelson, Rauhut, and Ward, Improved Bounds for Sparse Recovery from Subsampled Random Convolutions, §2.1, Definition 2.1, printed pp. 4–5

Foucart and Rauhut, A Mathematical Introduction to Compressive Sensing, §4.3, Definitions 4.17 and 4.21

Depends on

Used by

Dependency tree · two levels

30 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

Shahar Mendelson, Holger Rauhut, and Rachel Ward, Improved Bounds for Sparse Recovery from Subsampled Random Convolutions, §2.1 Definition 2.1 (standard reference, not scraped)
Simon Foucart and Holger Rauhut, A Mathematical Introduction to Compressive Sensing, §4.3 (standard reference, not scraped)