  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.
