Sampling and Recovery of Signals Over Product Cell Structures

We consider recovery of signals over product cell structures from subsampled observations. In particular, we consider product cell complexes, which can be factorized as the Cartesian product of two simplicial complexes. We focus on recovering edge and node signals from subsampled signals on the factor simplicial complexes. To do so, we first express bandlimited edge signals on the product cell complex as a direct sum of the Kronecker product of bandlimited edge and node signals on the factor simplicial complexes. Then, we propose a simple least squares solution for estimating the edge signal on the product complex. Next, we leverage the Helmholtz-Hodge decomposition on product spaces and propose a simple least squares estimator to recover node signals on the product cell complex. We evaluate the proposed method on both synthetic and real data.