Filters are fundamental in extracting information from data. For time series and image data that reside on Euclidean domains, filters are the crux of many signal processing and machine learning techniques, including convolutional neural networks. Increasingly, modern data also reside on networks and other irregular domains whose structure is better captured by a graph. To process and learn from such data, graph filters account for the structure of the underlying data domain. In this article, we provide a comprehensive overview of graph filters, including the different filtering categories, design strategies for each type, and trade-offs between different types of graph filters. We discuss how to extend graph filters into filter banks and graph neural networks to enhance the representational power; that is, to model a broader variety of signal classes, data patterns, and relationships. We also showcase the fundamental role of graph filters in signal processing and machine learning applications. Our aim is that this article provides a unifying framework for both beginner and experienced researchers, as well as a common understanding that promotes collaborations at the intersections of signal processing, machine learning, and application domains.

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Driven by the outstanding performance of neural networks in the structured euclidean domain, recent years have seen a surge of interest in developing neural networks for graphs and data supported on graphs. The graph is leveraged at each layer of the neural network as a parameterization to capture detail at the node level with a reduced number of parameters and computational complexity. Following this rationale, this paper puts forth a general framework that unifies state-of-the-art graph neural networks (GNNs) through the concept of EdgeNet. An EdgeNet is a GNN architecture that allows different nodes to use different parameters to weigh the information of different neighbors. By extrapolating this strategy to more iterations between neighboring nodes, the EdgeNet learns edge- and neighbor-dependent weights to capture local detail. This is a general linear and local operation that a node can perform and encompasses under one formulation all existing graph convolutional neural networks (GCNNs) as well as graph attention networks (GATs). In writing different GNN architectures with a common language, EdgeNets highlight specific architecture advantages and limitations, while providing guidelines to improve their capacity without compromising their local implementation. For instance, we show that GCNNs have a parameter sharing structure that induces permutation equivariance. This can be an advantage or a limitation, depending on the application. In cases where it is a limitation, we propose hybrid approaches and provide insights to develop several other solutions that promote parameter sharing without enforcing permutation equivariance. Another interesting conclusion is the unification of GCNNs and GATs - approaches that have been so far perceived as separate. In particular, we show that GATs are GCNNs on a graph that is learned from the features. This particularization opens the doors to develop alternative attention mechanisms for improving discriminatory power.

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Graph convolutional neural networks (GCNNs) learn compositional representations from network data by nesting linear graph convolutions into nonlinearities. In this work, we approach GCNNs from a state-space perspective revealing that the graph convolutional module is a minimalistic linear state-space model, in which the state update matrix is the graph shift operator. We show that this state update may be problematic because it is nonparametric, and depending on the graph spectrum it may explode or vanish. Therefore, the GCNN has to trade its degrees of freedom between extracting features from data and handling these instabilities. To improve such trade-off, we propose a novel family of nodal aggregation rules that aggregate node features within a layer in a nonlinear state-space parametric fashion allowing for a better trade-off. We develop two architectures within this family inspired by the recurrence with and without nodal gating mechanisms. The proposed solutions generalize the GCNN and provide an additional handle to control the state update and learn from the data. Numerical results on source localization and authorship attribution show the superiority of the nonlinear state-space generalization models over the baseline GCNN.@en

Convolutional neural networks (CNNs) restrict the, otherwise arbitrary, linear operation of neural networks to be a convolution with a bank of learned filters. This makes them suitable for learning tasks based on data that exhibit the regular structure of time signals and images. The use of convolutions, however, makes them unsuitable for processing data that do not exhibit such a regular structure. Graph signal processing (GSP) has emerged as a powerful alternative to process signals whose irregular structure can be described by a graph. Central to GSP is the notion of graph convolutional filters which can be used to define convolutional graph neural networks (GNNs). In this paper, we show that the graph convolution can be interpreted as either a diffusion or aggregation operation. When combined with nonlinear processing, these different interpretations lead to different generalizations which we term selection and aggregation GNNs. The selection GNN relies on linear combinations of signal diffusions at different resolutions combined with node-wise non-linearities. The aggregation GNN relies on linear combinations of neighborhood averages of different depth. Instead of node-wise nonlinearities, the nonlinearity in aggregation GNNs is pointwise on the different aggregation levels. Both of these models particularize to regular CNNs when applied to time signals but are different when applied to arbitrary graphs. Numerical evaluations show different levels of performance for selection and aggregation GNNs.

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