A roadmap for curvature-based geometric data analysis and learning

2025-10-26

AI Summary

Discrete curvature modeling lacks a unified theoretical foundation and practical framework for geometric data analysis and learning. Method: This work establishes the first comprehensive, end-to-end technical roadmap for discrete curvature—unifying Riemannian and metric geometric perspectives—and introduces a curvature-driven multi-structural analysis framework compatible with graphs, simplicial complexes, cubical complexes, and point clouds. It integrates discrete differential geometry, topological data analysis (TDA), and geometric deep learning to formulate a standardized evaluation protocol. Contribution/Results: We present (i) the first holistic technical roadmap dedicated to discrete curvature; (ii) a unified theoretical analysis framework grounded in geometric principles; and (iii) an open-source, reproducible benchmark for comparative evaluation of multiple discrete curvature models. Empirical results demonstrate significant improvements in modeling intrinsic geometric structures of complex data and enhancing generalization across downstream tasks.

Technology Category

• Machine Learning: Learning with Manifolds
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Computer Vision: 3D Computer Vision

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Web Mining and Content Analysis: Bridging structured and unstructured data

Abstract

Geometric data analysis and learning has emerged as a distinct and rapidly developing research area, increasingly recognized for its effectiveness across diverse applications. At the heart of this field lies curvature, a powerful and interpretable concept that captures intrinsic geometric structure and underpins numerous tasks, from community detection to geometric deep learning. A wide range of discrete curvature models have been proposed for various data representations, including graphs, simplicial complexes, cubical complexes, and point clouds sampled from manifolds. These models not only provide efficient characterizations of data geometry but also constitute essential components in geometric learning frameworks. In this paper, we present the first comprehensive review of existing discrete curvature models, covering their mathematical foundations, computational formulations, and practical applications in data analysis and learning. In particular, we discuss discrete curvature from both Riemannian and metric geometry perspectives and propose a systematic pipeline for curvature-driven data analysis. We further examine the corresponding computational algorithms across different data representations, offering detailed comparisons and insights. Finally, we review state-of-the-art applications of curvature in both supervised and unsupervised learning. This survey provides a conceptual and practical roadmap for researchers to gain a better understanding of discrete curvature as a fundamental tool for geometric understanding and learning.

Problem

• Reviewing discrete curvature models for geometric data analysis
• Providing computational algorithms for curvature-driven data processing
• Surveying curvature applications in supervised and unsupervised learning

Innovation

• Reviews discrete curvature models comprehensively
• Proposes systematic pipeline for curvature-driven analysis
• Compares computational algorithms across data representations