AATRN Watch Party: Discrete Ollivier-Ricci curvature for data visualization and analysis

When working with high-dimensional, noisy data, it can be difficult to accurately capture underlying geometric and topological features. Data points that appear close within the ambient space may be far with respect to geodesic distance; meanwhile, nonlinear dimension-reduction algorithms often fragment clusters of points that are in fact close to each other on the manifold. In this talk I’ll show how discrete curvature (specifically Ollivier-Ricci curvature) of a nearest-neighbor graph can be used to mitigate both of these problems. We’ll start by introducing and defining Ollivier-Ricci curvature (ORC). In our first paper, we develop an algorithm that uses ORC to prune “shortcut” edges. Pruning improves many downstream tasks such as persistent homology, geodesic-distance estimation, and nonlinear dimension reduction. In our second paper, we use ORC to define a metric that we put into the stochastic neighbor embedding (SNE) framework to produce visualizations that highlight cluster structure while simultaneously avoiding fragmentation. In both papers, we find that using ORC significantly improves our ability to recover manifold and cluster structure.