Recovering Manifold Structure Using Ollivier-Ricci Curvature

TL;DR

This paper presents ORC-ManL, an algorithm that uses Ollivier-Ricci curvature to prune noisy edges in nearest neighbor graphs, improving manifold learning and data analysis tasks.

Contribution

The paper introduces a novel curvature-based pruning method for nearest neighbor graphs, enhancing manifold learning and downstream data analysis performance.

Findings

Outperforms existing pruning methods in manifold learning tasks.

Significantly improves downstream geometric data analysis results.

Empirically supports theoretical convergence claims.

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.