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Graph Filters for Signal Processing and Machine Learning on Graphs

Elvin Isufi, Fernando Gama, David I. Shuman, Santiago Segarra

TL;DR

This article discusses how to extend graph filters into filter banks and graph neural networks to enhance the representational power, to model a broader variety of signal classes, data patterns, and relationships.

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 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.

Graph Filters for Signal Processing and Machine Learning on Graphs

TL;DR

This article discusses how to extend graph filters into filter banks and graph neural networks to enhance the representational power, to model a broader variety of signal classes, data patterns, and relationships.

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 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.
Paper Structure (44 sections, 6 theorems, 91 equations, 7 figures, 1 table, 1 algorithm)

This paper contains 44 sections, 6 theorems, 91 equations, 7 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Organization
  3. Graphs and Signals
  4. Graph Terminology
  5. Signals Defined on Graphs and Common Processing Tasks
  6. Graph Convolutional Filters
  7. Definition
  8. Properties
  9. Spectral Analysis
  10. Graph Fourier Transform
  11. Frequency Response
  12. Filter Design and Identification
  13. Operator Matching
  14. Data-driven
  15. Other Graph Filters
  16. ...and 29 more sections

Key Result

Proposition 1

Given the three following conditions: Then, $\mathbf{B} = \mathbf{H}(\mathbf{h}^*, \mathbf{S})$ where $\mathbf{h}^* = \mathbf{\Psi}^\dag \boldsymbol{\beta}$, $\mathbf{\Psi}$ is the Vandermonde matrix defined after eq:GFTfilter, and $(\cdot)^\dag$ denotes the pseudo-inverse.

Figures (7)

  • Figure 1: A roadmap of this article. Solid arrows prerequisite relationships between sections. For example, Sec. \ref{['sec:wavelets']} can be mostly understood without reading Sec. \ref{['sec:other']}, but not without reading Sec. \ref{['sec:filt_spect']}. The boxes for applications in signal processing and machine learning correspond to specific application examples we discuss in this article, and are not meant to be a comprehensive representation of all work that has been done in the field.
  • Figure 2: The graph convolutional filter as a shift register. Highlighted are the nodes that reach node $1$ on each consecutive shift; that is, the nodes $j$ whose signal value $x_j$ contributes to $[\mathbf{S}^k\mathbf{x}]_i$. The resulting summary of each communication $\mathbf{S}^{k}\mathbf{x}$ is correspondingly weighted by a filter parameter $h_{k}$. For each $k$, the parameter $h_k$ is the same for all nodes. In this example, $\mathbf{S}=\mathbf{L}_n$ and $\mathbf{H}(\mathbf{S})=1\mathbf{L}_n^0-1.5\mathbf{L}_n^1+1\mathbf{L}_n^2-0.25\mathbf{L}_n^3$ is a lowpass filter that smooths the input signal.
  • Figure 3: Discrete-time periodic signals as graph signals over a directed cycle graph. Each node $\mathcal{V}_c = \{1, \ldots, 6\}$ is a time instant with adjacencies captured in the matrix $\mathbf{A}_c$. The temporal signal forms the graph signal $\mathbf{x} = [x_1, \ldots, x_6]^{\mathsf{T}}$ and the shift $\mathbf{A}_c \mathbf{x}$ acts as a delay operation that moves the signal to the next time instant node.
  • Figure 4: The frequency response of the filter \ref{['eq:freqResponse']}, given in the black solid line, is completely characterized by the values of the filter parameters $\mathbf{h}$. Given a graph, this frequency response gets instantiated on the specific eigenvalues of that graph, determining the effect the filter will have on an input \ref{['eq:GFToutput']}.
  • Figure 5: An undecimated single-level four-channel graph filter bank. The signal is piecewise smooth with respect to the Stanford bunny graph bunny. As a result, the non-zero coefficients of the bandpass and highpass channels ($m=2,3,4$) cluster around the two discontinuities at the midsection and tail of the bunny. The synthesis filters $\{\mathsf{\tilde{g}}_m\}$ used here are the same as the analysis filters $\{\mathsf{\tilde{h}}_m\}$, although they do not need to be in general. The filters $\{\mathsf{\tilde{h}}_m\}$ chosen for this example with the design method of shuman2013spectrum satisfy the tight Parseval frame condition \ref{['Eq:Parseval']}, leading to perfect reconstruction.
  • ...and 2 more figures

Theorems & Definitions (6)

  • Proposition 1: Segarra2017-GraphFilterDesign
  • Proposition 2: Segarra2017-GraphFilterDesign
  • Proposition 3: Segarra2017-GraphFilterDesign
  • Proposition 4: Finite-time consensus sandryhaila2014finite
  • Proposition 5: gao2021stability
  • Proposition 6: hua2020online