Speeding Up Latent Variable Gaussian Graphical Model Estimation via Nonconvex Optimizations
This work addresses a computational bottleneck in statistical modeling for fields like genomics, offering a significant speed improvement over existing methods.
The paper tackles the slow estimation of latent variable Gaussian graphical models by proposing a nonconvex optimization method with an alternating gradient descent algorithm, achieving orders of magnitude faster computation than convex methods while proving linear convergence to optimal statistical precision.
We study the estimation of the latent variable Gaussian graphical model (LVGGM), where the precision matrix is the superposition of a sparse matrix and a low-rank matrix. In order to speed up the estimation of the sparse plus low-rank components, we propose a sparsity constrained maximum likelihood estimator based on matrix factorization, and an efficient alternating gradient descent algorithm with hard thresholding to solve it. Our algorithm is orders of magnitude faster than the convex relaxation based methods for LVGGM. In addition, we prove that our algorithm is guaranteed to linearly converge to the unknown sparse and low-rank components up to the optimal statistical precision. Experiments on both synthetic and genomic data demonstrate the superiority of our algorithm over the state-of-the-art algorithms and corroborate our theory.