Stratified Graph Spectra

TL;DR

This paper introduces a novel spectral analysis method for vector-valued graph signals, addressing limitations of traditional graph Fourier transforms and enhancing the profiling of graph learning models.

Contribution

It proposes a new stratified graph spectral transform that operates at hierarchical adjacency levels for better spectral characterization of vector signals.

Findings

Hierarchical spectral analysis provides deeper insights into graph signals.

The method aids in diagnosing and profiling graph learning models.

Traditional Fourier transforms are insufficient for vector-valued signals.

Abstract

In classic graph signal processing, given a real-valued graph signal, its graph Fourier transform is typically defined as the series of inner products between the signal and each eigenvector of the graph Laplacian. Unfortunately, this definition is not mathematically valid in the cases of vector-valued graph signals which however are typical operands in the state-of-the-art graph learning modeling and analyses. Seeking a generalized transformation decoding the magnitudes of eigencomponents from vector-valued signals is thus the main objective of this paper. Several attempts are explored, and also it is found that performing the transformation at hierarchical levels of adjacency help profile the spectral characteristics of signals more insightfully. The proposed methods are introduced as a new tool assisting on diagnosing and profiling behaviors of graph learning models.