Reviews: Speeding Up Latent Variable Gaussian Graphical Model Estimation via Nonconvex Optimization

The paper considers learning the dependency structure of Gaussian graphical models where some variables are latent. Directly applying the usual assumption of sparsity in the precision matrix is difficult because variables that appear correlated might actually both depend on a common latent variable. Previously, Chandrasekaran et al. proposed estimating the model structure by decomposing the full precision matrix into the sum of of a sparse matrix and a low-rank matrix. Likelihood is maximized while the components of the sparse matrix are penalized with an l1 regularizer and the low-rank matrix is penalized with a nuclear norm. Computing the proximal operator to update the low-rank component requires performing SVD in O(d 3) time at each iteration. The authors propose replacing the low-rank component with its Cholesky decomposition ZZ T and finding Z directly.