In this paper, we provide a framework based upon diffusion processes for finding meaningful geometric descriptions of data sets.
We show that eigenfunctions of Markov matrices can be used to construct coordinates called diffusion maps that generate efficient
representations of complex geometric structures. The associated family of diffusion distances, obtained by iterating the Markov
matrix, defines multiscale geometries that prove to be useful in the context of data parametrization and dimensionality reduction.
The proposed framework relates the spectral properties of Markov processes to their geometric counterparts and it unifies ideas
arising in a variety of contexts such as machine learning, spectral graph theory and eigenmap methods.