A roadmap for curvature-based geometric data analysis and learning

TL;DR

This paper provides a comprehensive review of discrete curvature models in geometric data analysis, highlighting their mathematical foundations, computational methods, and applications in learning tasks across various data representations.

Contribution

It is the first extensive survey that systematically covers discrete curvature models, their algorithms, and their role in geometric data analysis and learning frameworks.

Findings

Comparison of computational algorithms for different data representations

Insights into curvature's role in supervised and unsupervised learning

Systematic pipeline for curvature-driven data analysis

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.