Topological and Geometric Approaches to Stratification Learning

Given potentially high-dimensional point cloud data, a fundamental question is whether one can infer the structure of the underlying space from which the data are sampled. Manifold learning approaches typically assume that the data lie on or near a single low-dimensional manifold. In practice, however, many datasets exhibit substantially richer behavior, including mixed dimensionality, intersections, and singularities. To model such phenomena, we consider data sampled from a mixture of possibly intersecting manifolds and aim to recover the constituent strata pieces corresponding to manifolds of varying dimensions.

This task falls within the area of stratification learning, which we use here in a broad sense to denote an unsupervised, exploratory paradigm for decomposing data into disjoint subsets that capture meaningful geometric and topological structure. In this talk, we discuss topological and geometric approaches to stratification learning, emphasizing tools from local homology, sheaf theory, and discrete stratified Morse theory.