Recovering Manifold Structure Using Ollivier Ricci Curvature

Tristan Luca Saidi, Abigail Hickok, Andrew J. Blumberg

Keywords: Manifold Learning, Persistent Homology, Ollivier-Ricci Curvature, Pruning, Nearest-Neighbor Graphs

TL;DR: We present and test a theoretically grounded method that uses discrete graph curvature to prune nearest-neighbor graphs.

Abstract: We introduce ORC-ManL, a new algorithm to prune spurious edges from nearest neighbor graphs using a criterion based on Ollivier-Ricci curvature and estimated metric distortion. Our motivation comes from manifold learning: we show that when the data generating the nearest-neighbor graph consists of noisy samples from a low-dimensional manifold, edges that shortcut through the ambient space have more negative Ollivier-Ricci curvature than edges that lie along the data manifold. We demonstrate that our method outperforms alternative pruning methods and that it significantly improves performance on many downstream geometric data analysis tasks that use nearest neighbor graphs as input. Specifically, we evaluate on manifold learning, persistent homology, dimension estimation, and others. We also show that ORC-ManL can be used to improve clustering and manifold learning of single-cell RNA sequencing data. Finally, we provide empirical convergence experiments that support our theoretical findings.

Primary Area: learning on graphs and other geometries & topologies

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Submission Number: 3315