Certify a graphical-lasso solution from its optimality conditions

Returns the largest violation of the graphical-lasso stationarity (KKT) conditions. Because the objective is strictly convex, a value near zero certifies the global optimum directly from the estimand, with no appeal to any reference solver. For W = Theta^{-1} the subgradient conditions are

• diagonal, unpenalised: \(W_{ii} = S_{ii}\);

• diagonal, penalised: \(W_{ii} - S_{ii} = \rho_{ii}\);

• off-diagonal with \(\Theta_{ij} \neq 0\): \(W_{ij} - S_{ij} = \rho_{ij}\,\mathrm{sign}(\Theta_{ij})\);

• off-diagonal with \(\Theta_{ij} = 0\): \(|W_{ij} - S_{ij}| \le \rho_{ij}\).

Usage

glasso_kkt(
  x,
  S = NULL,
  rho = NULL,
  penalize_diagonal = NULL,
  zero = NULL,
  active_tol = 1e-08
)

Arguments

x

A glasso_result from glasso_fit(), or a precision matrix.

S

Covariance the model was fit to. Required only when x is a bare matrix; taken from the fit otherwise.

rho

Penalty (scalar or p x p matrix). Required only when x is a bare matrix.

penalize_diagonal

Logical, whether the diagonal was penalised. Required only when x is a bare matrix.

zero

Two-column matrix of hard-constrained (row, col) index pairs, as passed to glasso_fit(). Taken from the fit when x is a glasso_result. Constrained entries are excluded from the check: at a hard-constrained edge the inactive-edge inequality does not apply, because the equality constraint carries its own multiplier that absorbs the residual. Checking them anyway reports optimal fits as non-optimal.

active_tol

Magnitude above which an off-diagonal entry counts as active. Default 1e-8.

Value

A single non-negative number: the maximum absolute stationarity violation. Values near zero certify optimality.

See also

glasso_fit()

Examples

set.seed(1)
x <- matrix(rnorm(200 * 4), ncol = 4)
fit <- glasso_fit(cov(x), rho = 0.05)
glasso_kkt(fit)
#> [1] 2.220446e-16