  Machine learning frameworks such as graph neural networks typically rely on a
given, fixed graph to exploit relational inductive biases and thus effectively
learn from network data. However, when said graphs are (partially) unobserved,
noisy, or dynamic, the problem of inferring graph structure from data becomes
relevant. In this paper, we postulate a graph convolutional relationship
between the observed and latent graphs, and formulate the graph learning task
as a network inverse (deconvolution) problem. In lieu of
eigendecomposition-based spectral methods or iterative optimization solutions,
we unroll and truncate proximal gradient iterations to arrive at a
parameterized neural network architecture that we call a Graph Deconvolution
Network (GDN). GDNs can learn a distribution of graphs in a supervised fashion,
perform link prediction or edge-weight regression tasks by adapting the loss
function, and they are inherently inductive. We corroborate GDN's superior
graph recovery performance and its generalization to larger graphs using
synthetic data in supervised settings. Furthermore, we demonstrate the
robustness and representation power of GDNs on real world neuroimaging and
social network datasets.
