The Spectral Graph Wavelet Transform: Fundamental Theory and Fast Computation

Abstract

The spectral graph wavelet transform (SGWT) defines wavelet transforms appropriate for data defined on the vertices of a weighted graph. Weighted graphs provide an extremely flexible way to model the data domain for a large number of important applications (such as data defined on vertices of social networks, transportation networks, brain connectivity networks, point clouds, or irregularly sampled grids). The SGWT is based on the spectral decomposition of the $N\times N$ graph Laplacian matrix $\mathscr {L}$, where N is the number of vertices of the weighted graph. Its construction is specified by designing a real-valued function g which acts as a bandpass filter on the spectrum of $\mathscr {L}$, and is analogous to the Fourier transform of the “mother wavelet” for the continuous wavelet transform. The wavelet operators at scale s are then specified by $T_g^s = g(s\mathscr {L})$, and provide a mapping from the input data $f\in \mathbb {R}^N$ to the wavelet coefficients at scale s. The individual wavelets $\psi _{s,n}$ centered at vertex n, for scale s, are recovered by localizing these operators by applying them to a delta impulse, i.e. $\psi _{s,n} = T_g^s \delta _n$. The wavelet scales may be discretized to give a graph wavelet transform producing a finite number of coefficients. In this work we also describe a fast algorithm, based on Chebyshev polynomial approximation, which allows computation of the SGWT without needing to compute the full set of eigenvalues and eigenvectors of $\mathscr {L}$.