A roadmap for curvature-based geometric data analysis and learning

TL;DR

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.

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.

Figures (8)

• Figure 1: Computation of Forman-Ricci curvature for an edge $e$ in an unweighted simplicial complex. The computation involves identifying the faces of $e$, which are the vertices at the ends of $e$; the cofaces of $e$, which are the 2-simplices or triangles containing $e$; and parallel edges, defined as edges that share either a common face or a common coface with $e$, but not both.
• Figure 2: Computation of Ollivier--Ricci curvature for an edge in a complete graph with five vertices. The edge $e = (v_1, v_2)$ is selected for curvature computation. The corresponding neighborhoods are identified as $\mathcal{N}_{v_1} = \{v_2, v_3, v_4, v_5\}$ and $\mathcal{N}_{v_2} = \{v_1, v_3, v_4, v_5\}$. Discrete probability measures $m_1^\alpha$ and $m_2^\alpha$ are constructed using $\alpha = \frac{1}{2}$, such that each vertex retains half of the mass and distributes the remaining half uniformly across its neighbors (each neighbor receives $\frac{1}{8}$). The right side of the figure displays the optimal transport plan $\rho_{ij}$ from $m_1^\alpha$ to $m_2^\alpha$. The red dashed arrows represent the flow of probability mass between the two measures. The thickness of each arrow is proportional to the amount of mass transferred.
• Figure 3: Computation of Bakry--Émery curvature on graphs. The Bakry--Émery curvature $p_{BE}(v)$ at a vertex $v$ can be determined entirely based on the local structure of its punctured 2-ball $\dot{B}_2(v)$. The punctured 2-ball consists of vertices in its 1-neighborhood $S_1(v)$ and 2-neighborhood $S_2(v)$, and includes all edges within $S_1(v)$ as well as those connecting $S_1(v)$ to $S_2(v)$, but excludes edges between nodes in $S_2(v)$. On the left, Bakry-Émery curvature of the vertex $v$ is strictly negative due to the presence of multiple disconnected components in $\dot{B}_2(v)$, reflecting a structural bottleneck at $v$. On the right, we show configurations that result in exceptions to this property, that is $p_{BE}(v) \geq 0$ even though $\dot{B}_2(v)$ is disconnected. These cases are explicitly characterized in cushing2020bakry.
• Figure 4: Upper and lower bounds of sectional curvature on graphs. Sectional curvature $\rho(v_1, v_2, v_3)$ is defined on a triple of vertices and quantifies the extent to which their neighborhood resembles a tree-like or fully connected structure. The figure demonstrates the computation of sectional curvature $\rho$ on three graph configurations: a star graph, a path graph, and a complete graph. In each case, closed balls $B_{r_i}(v_i)$ are initially assigned radii based on the Gromov products. These balls are expanded simultaneously until a non-empty intersection is achieved. The sectional curvature of the three vertices is the minimum value of the scaling factor at which this occurs. In the star and path graphs (left), the closed balls intersect without expansion, yielding the lower bound $\rho = 1$. In contrast, for the complete graph (right), the initial balls of radius $1/2$ do not intersect, and must be expanded to radius $1$, yielding the upper bound $\rho = 2$.
• Figure 5: Computation of Menger-Ricci and Haantjes-Ricci curvature on graphs(a) Example computation of Menger-Ricci curvature for an edge. The curvature is computed by aggregating the Menger curvatures of all triangles adjacent to the edge. (b) Example computation of Haantjes curvature for a path in the graph, treating the path as a discrete analogue of an arc and the shortest path between its endpoints as a chord. (c) Computation of Haantjes-Ricci curvature for an edge. In this case, the curvature is obtained by aggregating Haantjes curvatures over all the paths connecting the endpoints of the edge.