High-Dimensional Topological Data Analysis

Modern data often come as point clouds embedded in high dimensional Euclidean spaces, or possibly more general metric spaces. They are usually not distributed uniformly, but lie around some highly nonlinear geometric structures with nontrivial topology. Topological data analysis (TDA) is an emerging field whose goal is to provide mathematical and algorithmic tools to understand the topological and geometric structure of data. This chapter provides a short introduction to this new field through a few selected topics. The focus is deliberately put on the mathematical foundations rather than specific applications, with a particular attention to stability results asserting the relevance of the topological information inferred from data.