Bayesian reconstruction of Cartesian product graph signals with general patterns of missing data

In this paper, we address the challenge of signal reconstruction on Cartesian product graphs, which frequently arises in applications where node-level data may be corrupted by equipment failure or difficult to reliably collect. A recent approach to Graph Signal Reconstruction (GSR) is to formulate it as a Bayesian inference problem, where the underlying signal is presumed to be smooth with respect to the graph topology. However, the resulting linear system derived from this model specification is often intractable to solve directly for general Cartesian product graphs. To overcome this limitation, we introduce two robust and efficient iterative procedures for computing the posterior mean. The first method is a stationary iterative technique based on matrix splitting, while the second employs a conjugate gradient approach with a symmetric graph-spectral preconditioner. We provide a theoretical and numerical comparison of the convergence behaviour of these algorithms, demonstrating that the optimal choice depends on the model hyperparameters. Finally, we propose a novel supervised learning-based technique, enhanced by an active learning strategy, for estimating the posterior marginal variance. This circumvents the need to to instantiate, invert or decompose large matrices directly in memory and results in an R2 statistic of over 0.95 on a synthetic test data set. This paper contributes to the development of advanced signal reconstruction techniques applicable in various domains, including large-scale sensor networks and social network analysis.