LEARNING GAUSSIAN GRAPHICAL MODELS WITH DIFFERING PAIRWISE SAMPLE SIZES

High-dimensional Gaussian graphical models have been a powerful tool for learning connections or interaction patterns among a large number of variables. While most prior work focuses on the case when all variables are measured simultaneously, we consider a practically common but new setting when no simultaneous measurement of all variables is available and there are differing numbers of measurements on each pair of nodes. This occurs in estimating gene expression networks from single-cell sequencing data, functional connectivity from neuronal recordings, and sensor networks. Incorporating a projection step in the estimation of the covariance matrix, we develop a convex method for learning the neighborhood of any node. We provide theoretical guarantees for the proposed method, suggesting that the exact recovery of the neighborhood of any node $a$ hinges on sufficient joint measurements for the pairs of nodes that involve $a$'s neighbor(s). We also present some consequences of the theory and numerical experiments on special graphs.