Natural Graph Wavelet Packet Dictionaries

TL;DR

This paper introduces novel multiscale basis transforms for graph signals that leverage natural distances between Laplacian eigenvectors, generalizing wavelet packet dictionaries without relying on eigenvalue ordering.

Contribution

It proposes a new class of basis dictionaries for graph signals based on natural eigenvector distances, extending classical wavelet packets to arbitrary graphs.

Findings

Effective approximation of graph signals on sunflower graphs.

Improved representation of street network signals.

Algorithms for constructing these basis dictionaries.

Abstract

We introduce a set of novel multiscale basis transforms for signals on graphs that utilize their "dual" domains by incorporating the "natural" distances between graph Laplacian eigenvectors, rather than simply using the eigenvalue ordering. These basis dictionaries can be seen as generalizations of the classical Shannon wavelet packet dictionary to arbitrary graphs, and do not rely on the frequency interpretation of Laplacian eigenvalues. We describe the algorithms (involving either vector rotations or orthogonalizations) to construct these basis dictionaries, use them to efficiently approximate graph signals through the best basis search, and demonstrate the strengths of these basis dictionaries for graph signals measured on sunflower graphs and street networks.