Decimated Framelet System on Graphs and Fast G-Framelet Transforms

TL;DR

This paper introduces decimated framelets, a multiscale graph data representation system that enables efficient analysis and processing at multiple resolutions, with applications in graph neural networks and traffic analysis.

Contribution

It proposes a novel decimated framelet system on graphs, along with a fast G-framelet transform algorithm with linear complexity, supporting multiscale graph data analysis.

Findings

Linear time complexity for decimated G-framelet transforms.

Effective multiresolution analysis demonstrated on real-world graph data.

Numerical verification on random graphs confirms theoretical properties.

Abstract

Graph representation learning has many real-world applications, from super-resolution imaging, 3D computer vision to drug repurposing, protein classification, social networks analysis. An adequate representation of graph data is vital to the learning performance of a statistical or machine learning model for graph-structured data. In this paper, we propose a novel multiscale representation system for graph data, called decimated framelets, which form a localized tight frame on the graph. The decimated framelet system allows storage of the graph data representation on a coarse-grained chain and processes the graph data at multi scales where at each scale, the data is stored at a subgraph. Based on this, we then establish decimated G-framelet transforms for the decomposition and reconstruction of the graph data at multi resolutions via a constructive data-driven filter bank. The graph framelets are built on a chain-based orthonormal basis that supports fast graph Fourier transforms. From this, we give a fast algorithm for the decimated G-framelet transforms, or FGT, that has linear computational complexity O(N) for a graph of size N. The theory of decimated framelets and FGT is verified with numerical examples for random graphs. The effectiveness is demonstrated by real-world applications, including multiresolution analysis for traffic network, and graph neural networks for graph classification tasks.