  As a tool for estimating networks in high dimensions, graphical models are
commonly applied to calcium imaging data to estimate functional neuronal
connectivity, i.e. relationships between the activities of neurons. However, in
many calcium imaging data sets, the full population of neurons is not recorded
simultaneously, but instead in partially overlapping blocks. This leads to the
Graph Quilting problem, as first introduced by (Vinci et.al. 2019), in which
the goal is to infer the structure of the full graph when only subsets of
features are jointly observed. In this paper, we study a novel two-step
approach to Graph Quilting, which first imputes the complete covariance matrix
using low-rank covariance completion techniques before estimating the graph
structure. We introduce three approaches to solve this problem: block singular
value decomposition, nuclear norm penalization, and non-convex low-rank
factorization. While prior works have studied low-rank matrix completion, we
address the challenges brought by the block-wise missingness and are the first
to investigate the problem in the context of graph learning. We discuss
theoretical properties of the two-step procedure, showing graph selection
consistency of one proposed approach by proving novel L infinity-norm error
bounds for matrix completion with block-missingness. We then investigate the
empirical performance of the proposed methods on simulations and on real-world
data examples, through which we show the efficacy of these methods for
estimating functional connectivity from calcium imaging data.
