# Fast Graph Fourier Transforms Based on Graph Symmetry and Bipartition

## TL;DR

This paper introduces fast algorithms for graph Fourier transforms leveraging graph symmetry and bipartition, significantly reducing computational costs for symmetric and structured graphs in applications like image processing and spectral clustering.

## Contribution

It develops novel methods using Haar units and graph topological symmetry to accelerate GFT computation, especially for symmetric and nearly regular graphs.

## Key findings

- Significant reduction in computation costs demonstrated.
- Effective for symmetric and structured graphs like line, cycle, and skeletal graphs.
- Applicable in video compression and human action analysis.

## Abstract

The graph Fourier transform (GFT) is an important tool for graph signal processing, with applications ranging from graph-based image processing to spectral clustering. However, unlike the discrete Fourier transform, the GFT typically does not have a fast algorithm. In this work, we develop new approaches to accelerate the GFT computation. In particular, we show that Haar units (Givens rotations with angle $\pi/4$) can be used to reduce GFT computation cost when the graph is bipartite or satisfies certain symmetry properties based on node pairing. We also propose a graph decomposition method based on graph topological symmetry, which allows us to identify and exploit butterfly structures in stages. This method is particularly useful for graphs that are nearly regular or have some specific structures, e.g., line graphs, cycle graphs, grid graphs, and human skeletal graphs. Though butterfly stages based on graph topological symmetry cannot be used for general graphs, they are useful in applications, including video compression and human action analysis, where symmetric graphs, such as symmetric line graphs and human skeletal graphs, are used. Our proposed fast GFT implementations are shown to reduce computation costs significantly, in terms of both number of operations and empirical runtimes.