AlgorithmAlgorithm: 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.

Entrywise-thresholded sample covariance

Definition

Input: observations $X_{1}, \dots, X_{n} \in R^{p}$ , with integers $n \geq 1$ , $p \geq 2$ , and a fixed threshold constant $γ_{τ} > 0$ determined by the directional sub-Gaussian tail parameter $τ$ .

Compute the sample-centered covariance ${\hat{Σ}}^{*}$ from Sample covariance estimators and squared operator loss and put $λ = γ_{τ} \sqrt{\log (p) / n}$ . Output the symmetric matrix $\hat{Σ}$ defined by ${\hat{σ}}_{j j} = {\hat{σ}}_{j j}^{*}, {\hat{σ}}_{j k} = {\hat{σ}}_{j k}^{*} 1 {∣ {\hat{σ}}_{j k}^{*} ∣ \geq λ} (j \neq k) .$ The diagonal is unchanged; an off-diagonal entry at threshold equality is retained. This procedure is a definition and makes no risk or positive-semidefiniteness guarantee.

Depends on

Sample covariance estimators and squared operator loss

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

T. Tony Cai and Harrison H. Zhou, Optimal Rates of Convergence for Sparse Covariance Matrix Estimation, Section 4, equation (28) (standard reference, not scraped)