Abstract
Spectral analysis of graphs has lead to powerful algorithms, for example in machine learning, in particular for regression, classification and clustering. Eigenfunctions of the Laplacian on a graph are a natural basis for analyzing functions on a graph. In this talk we discuss a new flexible set of basis functions, called Diffusion Wavelets, that allow for a multiscale analysis of functions on a graph, very much in the same way classical wavelets perform a multiscale analysis in Euclidean spaces. They allow efficient, representation, compression, denoising of functions on the graph, and are very well-suited for statistical learning tasks, as well as unsupervised algorithms. They are also associated with a multiscale decomposition of the graph, which has applications by itself, for example to the analysis of time series of graphs, and for defining families of new robust distances between graphs. We will discuss this construction with several examples, going from signal processing on manifolds and graphs, to some applications to clustering and learning.