Geometric Methods for Statistical Learning and Data in High-Dimensions

We discuss a family of ideas, algorithms, and results for learning from high-dimensional data. These methods rely on the idea that complex high-dimensional data has geometric structures, often low-dimensional, that, once discovered, assist in a variety of statistical learning tasks, as well as in tasks such as data visualization. We discuss various realizations of these ideas, from manifold learning and dimension reduction techniques to new techniques based on suitable multiscale geometric decompositions of the data. We will then discuss how these multiscale decompositions may be used to solve various tasks, from dictionary learning to classification, to the construction of probabilistic models for the data, to approximation of high-dimensional stochastic systems.