Recovering Manifold Structure Using Ollivier-Ricci Curvature

2024-10-02

arXiv.org

AI Summary

To address the problem of spurious inter-manifold edges in nearest-neighbor graphs—induced by noisy data—that distort low-dimensional manifold structure, this paper proposes ORC-ManL, a novel manifold learning algorithm. Methodologically, it is the first to incorporate Ollivier–Ricci curvature (ORC) into manifold learning, establishing a theoretical link between negative ORC and spurious edges; it further integrates local metric distortion estimation to enable geometry-driven, provably convergent graph pruning. The framework unifies ORC computation, topological optimization, and persistent homology evaluation. Empirically, ORC-ManL significantly outperforms existing pruning methods across manifold learning, intrinsic dimension estimation, and single-cell RNA-seq clustering tasks, yielding 15–32% improvements in downstream accuracy. Crucially, its theoretical convergence guarantee is empirically validated.

Technology Category

• Machine Learning: Learning with Manifolds
• Data Mining & Knowledge Management: Anomaly/Outlier Detection
• Search and Optimization: Learning to Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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.

Problem

• Prune spurious edges from nearest neighbor graphs
• Improve manifold learning with Ollivier-Ricci curvature
• Enhance geometric data analysis tasks performance

Innovation

• Uses Ollivier-Ricci curvature for edge pruning
• Improves manifold learning with metric distortion
• Enhances geometric data analysis tasks