Peng and Shen study Gaussian graphical models in which the precision matrix decomposes as a sparse matrix minus a symmetric, possibly indefinite, low-rank matrix. This structure arises when latent-variable marginalization is combined with conditioning on linear measurements. Their proposed convex estimator replaces the sample covariance with a centered Gaussian measure-transformed covariance and penalizes the low-rank component using the nuclear norm of a congruence transform.
Under restricted invertibility and irrepresentability conditions on the Fisher information, the authors prove a primal–dual witness theorem: the raw convex optimizer is unique, has no false edges, recovers the signs of all sufficiently strong edges, and exactly recovers the true rank and inertia. These conditions reduce to the standard graphical-lasso conditions when the low-rank part is absent.
The paper also derives information-theoretic limits. Inertia recovery requires the sample size to scale at least as p divided by the square of the smallest nonzero eigenvalue modulus of the low-rank part, and the spectral beta-min condition is necessary up to a logarithmic factor. Frobenius estimation attains the minimax rate up to a factor involving incoherence, degree, and rank. Under Huber contamination, the authors give a sharp bias bound for the measure-transformed functional, and this bias vanishes for remote outliers. A linearized ADMM with closed-form updates provides a globally convergent algorithm.