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.

The positive-semidefinite Nyström approximation

Definition

Let $F \in {R, C}$ , let $A \in F^{n \times n}$ be Hermitian positive semidefinite with $n \geq 1$ , and let $S \in F^{n \times s}$ with $s \geq 0$ be any finite sketch matrix. Put $W = S^{*} A S \in F^{s \times s}$ . The Nyström approximation associated with $(A, S)$ is ${\hat{A}}_{n y s} = A S W^{†} S^{*} A$ .

The Moore-Penrose inverse is defined for every finite matrix, including singular $W$ , and is unique by the dependencies. No rank, sampling, or probabilistic condition is part of this definition.

Depends on

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

Per-Gunnar Martinsson and Joel A. Tropp, Randomized Numerical Linear Algebra: Foundations & Algorithms, §14.2 (standard reference, not scraped)
Alex Gittens and Michael W. Mahoney, Revisiting the Nyström Method, §4.1 (standard reference, not scraped)