Diffusion wavelets are a fast multiscale framework for the analysis of functions on discrete (or discretized continuous) structures like graphs, manifolds, and point clouds in Euclidean space.
Diffusion wavelets are an extension of classical wavelet theory from harmonic analysis.
Unlike classical wavelets whose basis functions are predetermined, diffusion wavelets are adapted to the geometry of a given diffusion operator
(e.g., a heat kernel or a random walk).
Moreover, the diffusion wavelet basis functions are constructed by dilation using the dyadic powers (powers of two) of
diffusion over the space and propagate local relationships in the function throughout the space until they become global.
decrease (i.e., its spectrum decays), then these higher powers become compressible.
From these decaying dyadic powers of
Diffusion wavelets were first introduced in 2004 by Ronald Coifman and Mauro Maggioni at Yale University.
[1] This algorithm constructs the scaling basis functions and the wavelet basis functions along with the representations of the diffusion operator
and the wavelet basis functions at scale
denotes the matrix representation of the scaling basis
denotes the matrix represents of the operator
and the range is represented with respect to the basis
[2] Diffusion wavelets are of general interest in mathematics.
Specifically, they allow for the direct calculation of the Green′s function and the inverse graph Laplacian.
Diffusion wavelets have been used extensively in computer science, especially in machine learning.