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.

翻译：图神经网络（GNNs）已成为处理非欧几里得数据集学习问题的著名替代方案。然而，尽管大多数研究假设图是完全已知的，但观测到的拓扑结构容易受到来自观测噪声、图学习限制或对抗性攻击的误差影响。如果忽略这些扰动，可能会严重阻碍GNN的性能。为解决这一局限，本文提出了一种鲁棒性GNN实现，该实现显式考虑了观测拓扑中存在的扰动。对于涉及GNN的任何任务，我们的核心思想是：i) 不仅针对GNN的可学习参数，还针对真实图求解优化问题，ii) 在拟合代价中增加一项来表征图上的差异。具体而言，我们考虑基于图滤波器的卷积GNN，并采用交替优化方法，通过梯度下降与投影近端更新相结合来处理（非可微且带约束的）优化问题。所得到的算法不局限于特定类型的图，并且能够融入关于扰动的先验信息。最后，通过多项数值实验评估了所提出方法的性能。