Detecting singularities from non-manifold data with HADES

High-dimensional data often exhibit low-dimensional structure, and a geometer's first intuition is to model the data using smooth manifolds. However this common assumption, called the "Manifold Hypothesis", is rarely verified to be true. In this talk I will introduce HADES, an unsupervised algorithm to detect data points that show non-manifold behaviour (i.e. singularities in data).

This algorithm uses simple statistical methods that makes it much faster than existing topology-based alternatives. Using tools from differential geometry and optimal transport theory, we prove that HADES correctly detects singularities with high probability when the data sample lives on a transverse intersection of equidimensional manifolds. In computational experiments, HADES recovers singularities in synthetically generated data, branching points in road network data, intersection rings in molecular conformation space, and anomalies in image data.