Taming Latent Ill-Conditioning in Gaussian Graphical Model Estimation

Latent-variable Gaussian graphical models separate a sparse conditional graph from the collective influence of unobserved variables. Although low-rank factorization makes this decomposition convenient to optimize, latent effects of unequal strength can slow gradient descent even when the observed precision matrix is well-conditioned. To address this difficulty, we combine scaled gradient descent with symmetric hard thresholding and a joint feasibility and descent line search. At the correctly specified rank, we establish a local linear convergence bound for the separate sparse and low-rank matrix errors, up to statistical accuracy. The contraction factor and statistical-error multiplier are independent of the latent eigenvalues, with precision conditioning and sparse–low-rank transversality held fixed. The analysis connects an exact cancellation in the scaled gradient to the geometry of fixed-rank matrices, allowing both components to be controlled along the same trajectory. We further provide sufficient conditions under which a thresholded spectral initializer identifies the support and enters the convergence neighborhood. In simulations, ScaledGD maintains similar late-stage convergence rates as latent conditioning worsens while gradient descent slows, and runs faster than AltGD in the sampled-data comparisons.