Graph Filters for Signal Processing and Machine Learning on Graphs

Abstract

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 serves the dual purpose of providing a unifying framework for both beginner and experienced researchers, as well as a common understanding that promotes collaborations between signal processing, machine learning, and application domains.

Voxel-Wise Brain Graphs From Diffusion MRI: Intrinsic Eigenspace Dimensionality and Application to Functional MRI

“…Lastly, the exquisite voxel-wise scale of the proposed graphs can enable assessing the extent to which brain structural-functional relations hold at spatially finer mesoscales [30]; e.g. by using graph Slepians [56]- [58] or variants of localized graph filter banks [59], [60] and spectral transforms [61]- [63], focus can be placed on a particular subset of nodes, thus, providing a finer level of analytical resolution than that provided by conventional region-wise graphs.…”

Voxel-Wise Brain Graphs from Diffusion MRI: Intrinsic Eigenspace Dimensionality and Application to Functional MRI

“…Lastly, the exquisite voxel-wise scale of the proposed graphs can enable assessing the extent to which brain structural-functional relations hold at spatially finer mesoscales (Mansour-L et al, 2021); e.g. by using graph Slepians (Van De Ville et al, 2017; Petrovic et al, 2019; Georgiadis et al, 2021) or variants of localized graph filter banks (Shuman, 2020; Isufi et al, 2022) and spectral transforms (Ghandehari et al, 2021; de Loynes et al, 2022; Tay, 2022), focus can be placed on a particular subset of nodes, thus, providing a finer level of analytical resolution than that provided by conventional region-wise graphs.…”

Spectral Representation of EEG Data using Learned Graphs with Application to Motor Imagery Decoding

“…as in (Behjat et al, 2015, 2021), as this can be necessary in deriving compact, interpretable EEG-based clinical biomarkers (Mussigmann et al, 2022). The challenge to this aim would be to use an appropriate choice of graph filter banks Shuman (2020); Isufi et al (2022) based on the graph spectra (Shuman et al, 2015) and the energy spectral density (Behjat et al, 2016) of EEG maps on learned graphs.…”