Recovery of Signals on a Simplicial Complex From Subsampled Neighborhood Aggregations

In this work, we focus on recovering signals over simplicial complexes from subsampled observations. In particular, we subsample a simplicial signal of a certain order and focus on recovering multi-order bandlimited simplicial signals of one order higher and one order lower, wherein the observations are collected using a neighborhood aggregation sampling mechanism. To do so, we assume that the simplicial signal admits the Hodge decomposition that relates simplicial signals of different orders. Next, we propose a simple least squares estimator for recovery. We also provide theoretical conditions on the number of aggregations and size of the sampling set required for faithful reconstruction as a function of the bandwidth of simplicial signals to be recovered. Numerical experiments are provided to show the effectiveness of the proposed method.