Signal Processing and Machine Learning on Graphs, Part I of 2

The field of graph signal processing extends classical signal processing tools to signals (data) with an irregular structure that can be characterized by means of a graph (e.g., network data). One of the building blocks of this field are graph filters, direct analogues of time-domain filters, but intended for signals defined on graphs. In this tutorial, we introduce the field of graph signal processing and specifically give an overview of the graph filtering problem. We look at the family of finite impulse response (FIR) and infinite impulse response (IIR) graph filters and show how they can be implemented in a distributed manner. To further limit the communication and computational complexity of such a distributed implementation, we also generalize the state-of-the-art distributed graph filters to filters whose weights show a dependency on the nodes sharing information. These so-called edge-variant graph filters yield significant benefits in terms of filter order reduction and can be used for solving specific distributed optimization problems with extremely fast convergence. Finally, we will overview how graph filters can be used in deep learning applications involving data sets with an irregular structure. Different types of graph filters can be used in the convolution step of graph convolutional networks leading to different trade-offs in performance and complexity. The numerical results presented in this talk illustrate the potential of graph filters in signal processing and machine learning.