Stratified Graph Spectra
This work provides a new tool for diagnosing and profiling behaviors in graph learning models, addressing a specific mathematical gap for researchers in graph signal processing and machine learning.
The paper tackles the problem of defining a valid graph Fourier transform for vector-valued graph signals, which is not addressed by classic graph signal processing, and proposes methods that perform transformations at hierarchical adjacency levels to better profile spectral characteristics.
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.