Diffusion Maps for Dimensionality Reduction

In this thesis we present the diffusion maps, a framework based on diffusion processes for finding meaningful geometric descriptions of data sets. A diffusion process can be described via an iterative application of the heat kernel which has two main characteristics: it satisfies a Markov semigroup property and its level sets encode all geometric features of the space. This process, well known in regular manifolds, has been extended to general data set by Coifman and Lafon. They define a diffusion kernel starting from the geometric properties of the data and their density properties. This kernel will be a compact operator, and the projection on its eigenvectors at different instant of time, provides a family of embeddings of a dataset into a suitable Euclidean space. The projection on the first eigenvectors, naturally leads to a dimensionality reduction algorithm. Numerical implementation is provided on different data set.