ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)

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Increasing λ creates an exact zero edge in a two-variable graphical lasso

Example

Take observations $x_{1} = (1, 1)^{T}$ and $x_{2} = (- 1, - 1)^{T}$ , so their centered sample covariance is $S = (\begin{array}{cc} 1 & 1 \\ 1 & 1 \end{array})$ . Under the all-entry penalty objective, every $λ \geq 1$ gives the unique maximizer $Θ = (1 + λ)^{- 1} I_{2}$ with $Θ_{12} = 0$ , while for $0 < λ < 1$ every optimizer has $Θ_{12} \neq 0$ . In particular, $λ = 2$ yields a zero edge and $λ = \frac{1}{2}$ yields a nonzero edge.

Example references

Friedman, Hastie and Tibshirani, §2

The Elements of Statistical Learning, §17.3.2

Facts & Assumptions

Given: The two observations and the all-entry graphical-lasso convention in the Example.

[F1]

The centered sample covariance and objective use $∥ Θ ∥_{1} = \sum_{i, j} ∣ Θ_{i j} ∣$ , including diagonal entries. Gaussian sparse-precision objective with the all-entry penalty.

[F2]

For $S ⪰ 0$ and $λ > 0$ , the objective has a unique maximizer characterized by the symmetric entrywise KKT condition; at a zero off-diagonal entry the subgradient lies in $[- 1, 1]$ . The all-entry graphical-lasso objective has a unique KKT optimizer.

Verification

technique · direct

1.1givenF1algebra

The sample mean is zero, and $S = \frac{1}{2} (x_{1} x_{1}^{T} + x_{2} x_{2}^{T}) = (\begin{array}{cc} 1 & 1 \\ 1 & 1 \end{array})$ . This matrix is positive semidefinite because $u^{T} S u = (u_{1} + u_{2})^{2} \geq 0$ .

2.1F2step 1.1algebra

Let $λ \geq 1$ , $a = (1 + λ)^{- 1}$ , $Θ = a I_{2}$ , and let $Γ$ have diagonal entries $1$ and off-diagonal entries $- 1 / λ$ . Then $Θ ≻ 0$ , $Γ_{12} \in [- 1, 1]$ , and $Θ^{- 1} - S - λ Γ = 0$ : the diagonal entries are $(1 + λ) - 1 - λ = 0$ and the off-diagonal entries are $0 - 1 - λ (- 1 / λ) = 0$ . By [F2], this is the unique maximizer, with zero off-diagonal edge.

3.1givenF2step 1.1step 2.1algebra∎

Let $0 < λ < 1$ . If an optimizer had $Θ_{12} = 0$ , its off-diagonal KKT equation from [F2] would require $- 1 - λ Γ_{12} = 0$ for some $Γ_{12} \in [- 1, 1]$ , hence $Γ_{12} = - 1 / λ < - 1$ , a contradiction. Existence in [F2] therefore forces every optimizer to have $Θ_{12} \neq 0$ . Taking $λ = 2$ and $λ = \frac{1}{2}$ gives the stated two cases.

Depends on

Used by

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Sources

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