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Explore how discrete Ollivier-Ricci curvature can enhance geometric data analysis and visualization in this 52-minute conference talk. Learn the fundamental concepts of Ollivier-Ricci curvature (ORC) and discover how it addresses common challenges in high-dimensional, noisy data analysis where ambient space proximity doesn't reflect true geodesic distances and nonlinear dimension-reduction algorithms fragment connected clusters. Examine a novel algorithm that leverages ORC to prune "shortcut" edges in nearest-neighbor graphs, improving downstream tasks including persistent homology, geodesic-distance estimation, and nonlinear dimension reduction. Investigate how ORC can define metrics within the stochastic neighbor embedding (SNE) framework to create visualizations that emphasize cluster structure while preventing fragmentation. Understand the mathematical foundations of discrete curvature and see practical applications demonstrating significant improvements in recovering manifold and cluster structure from complex datasets.
