Graph quilting: graphical model selection from partially observed covariances

Abstract

We investigate the problem of conditional dependence graph estimation when several pairs of nodes have no joint observation. For these pairs even the simplest metric of covariability, the sample covariance, is unavailable. This problem arises, for instance, in calcium imaging recordings where the activities of a large population of neurons are typically observed by recording from smaller subsets of cells at once, and several pairs of cells are never recorded simultaneously. With no additional assumption, the unavailability of parts of the covariance matrix translates into the unidentifiability of the precision matrix that, in the Gaussian graphical model setting, specifies the graph. Recovering a conditional dependence graph in such settings is fundamentally an extremely hard challenge, because it requires to infer conditional dependences between network nodes with no empirical evidence of their covariability. We call this challenge the "graph quilting problem". We demonstrate that, under mild conditions, it is possible to correctly identify not only the edges connecting the observed pairs of nodes, but also a superset of those connecting the variables that are never observed jointly. We propose an 1 regularized graph estimator based on a partially observed sample covariance matrix and establish its rates of convergence in high-dimensions. We finally present a simulation study and the analysis of calcium imaging data of ten thousand neurons in mouse visual cortex.

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

“…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.…”

Gaussian Graphical Model Selection for Huge Data via Minipatch Learning

“…Concretely, suppose that we take a random minipatch that consists of a subset of nodes from the original data set and try to estimate the conditional dependence graph between these nodes by fitting a graph estimator to this minipatch. Then, the remaining nodes outside of this minipatch effectively become latent unobserved nodes with respect to the minipatch, thus inducing many false positive edges of small magnitude in this subgraph estimate (Vinci et al, 2019). Estimating graphs for each minipatch thus leads to the well-known latent variable graphical model problem (Chandrasekaran et al, 2012).…”

“…In neuroscience, Gaussian graphical models are often used to learn the functional connectivity between brain regions or neurons (Friston, 2011;Yatsenko et al, 2015;Chang et al, 2019). In fMRI data, the number of brain regions and hence graph nodes can be on the order of tens-to hundreds-of-thousands (Friston, 2011) whereas in calcium imaging, the number of neurons or graph nodes in some newer technologies can measure in the tens-of-thousands (Stringer and Pachitariu, 2019;Vinci et al, 2019).…”

The why, how, and when of representations for complex systems

“…Consequently, tremendous effort focuses on network reconstruction from data, i.e. inferring the interactions from indirect measurements [138,203,4]. Temporal information such as fluctuations in RNA counts [186] and spatial information such as co-localization [122] can be used to reconstruct the network of interactions.…”