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.

Gaussian sparse-precision objective with the all-entry penalty

Definition

For $N, p \geq 1$ and data $x_{1}, \dots, x_{N} \in R^{p}$ , put $\overset{ˉ}{x} = N^{- 1} \sum_{r = 1}^{N} x_{r}$ and $S = N^{- 1} \sum_{r = 1}^{N} (x_{r} - \overset{ˉ}{x}) (x_{r} - \overset{ˉ}{x})^{T}$ . For any symmetric positive-semidefinite $S$ and any $Θ \in S_{+ +}^{p}$ , set $ℓ_{S} (Θ) = \log \det Θ - tr (S Θ)$ , where $S_{+ +}^{p}$ is the cone of symmetric positive-definite $p \times p$ matrices. Define $∥ Θ ∥_{1} = \sum_{i = 1}^{p} \sum_{j = 1}^{p} ∣ Θ_{i j} ∣$ , including diagonal entries, and for $λ > 0$ define $F_{S, λ} (Θ) = ℓ_{S} (Θ) - λ ∥ Θ ∥_{1}$ . The off-diagonal-only convention $F_{S, λ}^{o f f} (Θ) = ℓ_{S} (Θ) - λ \sum_{i \neq j} ∣ Θ_{i j} ∣$ is a separate objective.

Definition references

The Elements of Statistical Learning, §17.3.1–§17.3.2

Friedman, Hastie and Tibshirani, §2

Used by

Dependency tree · 0 levels

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

Sources

The Elements of Statistical Learning, second edition (standard reference, not scraped)
Friedman, Hastie and Tibshirani, Sparse inverse covariance estimation with the lasso (standard reference, not scraped)