Speaker: Matthew Hirn (Yale University)

Title: Diffusion maps for changing data


Much of the data collected today is massive and high dimensional, yet hidden within is a low dimensional structure that is key to understanding it. As such, recently there has been a large class of research that utilizes nonlinear mappings into low dimensional spaces in order to organize high dimensional data according to its intrinsic geometry. Examples include, but are not limited to, locally linear embedding (LLE), ISOMAP, Hessian LLE, Laplacian eigenmaps, and diffusion maps. The type of question we shall ask in this talk is the following: if my data is in some way dynamic, either evolving over time or changing depending on some set of input parameters, how do these low dimensional embeddings behave? Is there a way to go between the embeddings, or better still, track the evolution of the data in its intrinsic geometry? Can we understand the global behavior of the data in a concise way? Focusing on the diffusion maps framework, we shall address these questions and a few others. We will begin with a review the original work on diffusion maps by Coifman and Lafon, and then present some current theoretical results. Various synthetic and real world examples will be presented to illustrate these ideas in practice, including examples taken from image analysis and dynamical systems. Parts of this talk are based on joint work with Ronald Coifman, Simon Adar, Yoel Shkolnisky, Eyal Ben Dor, and Roy Lederman.