  Graph Neural Networks (GNNs) have emerged as a notorious alternative to
address learning problems dealing with non-Euclidean datasets. However,
although most works assume that the graph is perfectly known, the observed
topology is prone to errors stemming from observational noise, graph-learning
limitations, or adversarial attacks. If ignored, these perturbations may
drastically hinder the performance of GNNs. To address this limitation, this
work proposes a robust implementation of GNNs that explicitly accounts for the
presence of perturbations in the observed topology. For any task involving
GNNs, our core idea is to i) solve an optimization problem not only over the
learnable parameters of the GNN but also over the true graph, and ii) augment
the fitting cost with a term accounting for discrepancies on the graph.
Specifically, we consider a convolutional GNN based on graph filters and follow
an alternating optimization approach to handle the (non-differentiable and
constrained) optimization problem by combining gradient descent and projected
proximal updates. The resulting algorithm is not limited to a particular type
of graph and is amenable to incorporating prior information about the
perturbations. Finally, we assess the performance of the proposed method
through several numerical experiments.
