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As a tool for estimating networks in high dimensions, graphical models arecommonly applied to calcium imaging data to estimate functional neuronalconnectivity, i.e. relationships between the activities of neurons. However, inmany calcium imaging data sets, the full population of neurons is not recordedsimultaneously, but instead in partially overlapping blocks. This leads to theGraph Quilting problem, as first introduced by (Vinci et.al. 2019), in whichthe goal is to infer the structure of the full graph when only subsets offeatures are jointly observed. In this paper, we study a novel two-stepapproach to Graph Quilting, which first imputes the complete covariance matrixusing low-rank covariance completion techniques before estimating the graphstructure. We introduce three approaches to solve this problem: block singularvalue decomposition, nuclear norm penalization, and non-convex low-rankfactorization. While prior works have studied low-rank matrix completion, weaddress the challenges brought by the block-wise missingness and are the firstto investigate the problem in the context of graph learning. We discusstheoretical properties of the two-step procedure, showing graph selectionconsistency of one proposed approach by proving novel L infinity-norm errorbounds for matrix completion with block-missingness. We then investigate theempirical performance of the proposed methods on simulations and on real-worlddata examples, through which we show the efficacy of these methods forestimating functional connectivity from calcium imaging data.

Low-Rank Covariance Completion for Graph Quilting with Applications to Functional Connectivity