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.

翻译：作为高维网络估计的工具，图模型常被应用于钙成像数据以估计功能性神经元连接性，即神经元活动之间的关系。然而，在许多钙成像数据集中，神经元整体并非被同时记录，而是以部分重叠的区块形式记录。这导致了图拼接问题（由 Vinci 等人于 2019 年首次提出），其目标是在仅联合观测到特征子集的情况下推断完整图的结构。本文研究了一种新颖的两步图拼接方法：首先利用低秩协方差补全技术补全协方差矩阵，随后估计图结构。我们提出了三种解决该问题的方法：区块奇异值分解、核范数惩罚以及非凸低秩分解。尽管已有研究探讨过低秩矩阵补全，但我们针对区块缺失带来的挑战进行了深入分析，并首次在图学习背景下研究该问题。我们讨论了两步程序的理论性质，通过证明区块缺失矩阵补全在 L∞ 范数误差界上的新结果，展示了其中一种方法在图选择一致性上的理论保证。最后，我们通过仿真实验和真实世界数据案例评估了所提出方法的实证性能，结果表明这些方法在钙成像数据的功能连接性估计中具有显著效果。