Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 205 / 2 / b

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- Solution b

Solution

The differential of the log-determinant is

d lo g det Ω = Tr (Ω^{- 1} d Ω)

. The subdifferential of the entrywise

ℓ^{1}

norm consists of symmetric matrices

Z

with

Z_{ij} = sgn (Ω_{ij}) if Ω_{ij} \neq = 0, Z_{ij} \in [- 1, 1] if Ω_{ij} = 0.

The Karush-Kuhn-Tucker conditions for the Graphical Lasso are therefore

- Ω^{- 1} + S + λ Z = 0, Z \in \partial ∥ Ω ∥_{1, entry} .

Because

- lo g det Ω

is strictly convex on the positive-definite matrices, these conditions characterize the unique minimizer whenever it exists.

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