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.

Signal processing (SP) excels at analyzing, processing, and inferring information defined over regular (first continuous, later discrete) domains such as time or space. Indeed, the last 75 years have shown how SP has made an impact in areas such as communications, acoustics, sensing, image processing, and control, to name a few. With the digitalization of the modern world and the increasing pervasiveness of data-collection mechanisms, information of interest in current applications oftentimes arises in non-Euclidean, irregular domains. Graph SP (GSP) generalizes SP tasks to signals living on non-Euclidean domains whose structure can be captured by a weighted graph. Graphs are versatile, able to model irregular interactions, easy to interpret, and endowed with a corpus of mathematical results, rendering them natural candidates to serve as the basis for a theory of processing signals in more irregular domains.