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Graphical model selection is a seemingly impossible task when many pairs ofvariables are never jointly observed; this requires inference of conditionaldependencies with no observations of corresponding marginal dependencies. Thisunder-explored statistical problem arises in neuroimaging, for example, whendifferent partially overlapping subsets of neurons are recorded innon-simultaneous sessions. We call this statistical challenge the "GraphQuilting" problem. We study this problem in the context of sparse inversecovariance learning, and focus on Gaussian graphical models where we show thatmissing parts of the covariance matrix yields an unidentifiable precisionmatrix specifying the graph. Nonetheless, we show that, under mild conditions,it is possible to correctly identify edges connecting the observed pairs ofnodes. Additionally, we show that we can recover a minimal superset of edgesconnecting variables that are never jointly observed. Thus, one can inferconditional relationships even when marginal relationships are unobserved, asurprising result! To accomplish this, we propose an $\ell_1$-regularizedpartially observed likelihood-based graph estimator and provide performanceguarantees in population and in high-dimensional finite-sample settings. Weillustrate our approach using synthetic data, as well as for learningfunctional neural connectivity from calcium imaging data.

Graph quilting: graphical model selection from partially observed covariances