Curvature Information Collection

• Curvature information collection is the systematic extraction and interpretation of local and global curvature features from discretized structures like point clouds, graphs, and simplicial complexes.
• It employs robust algorithmic pipelines—encompassing surface sampling, metric extraction, and holonomy computation—to yield curvature estimates with provable convergence and stability.
• Applications span geometric deep learning, network analysis, and discrete gravity, where curvature serves as a critical feature for enhancing modeling, anomaly detection, and optimization.

Curvature information collection refers to the systematic extraction, computation, and interpretation of local and global geometric curvature features from discretely sampled or combinatorial data structures, including point clouds, simplicial complexes, graphs, and polyhedral surfaces. This process is foundational across discrete differential geometry, computational topology, geometric learning, and applied fields such as computer vision, physics, and network analysis, where quantifying and leveraging curvature enables geometry-aware modeling, analysis, and inference. Recent advances provide algorithmic pipelines for collecting and aggregating curvature data on a wide variety of discrete representations, with rigorous mathematical underpinnings and validated performance in both theoretical and applied contexts.

1. Geometric and Mathematical Foundations

Curvature captures how a geometric object departs from being flat or straight, with precise local invariants in the smooth setting (e.g., Gaussian, mean, and sectional curvature for manifolds; Ricci curvature for Riemannian spaces; turning angle for planar polygons). In discrete geometry and combinatorics, analogous notions are developed to retain geometric and topological meaning on structures such as:

• Pointwise curvature for surfaces: Via combinatorial angle defects or deficits at vertices in polyhedral surfaces (Izmestiev, 13 Feb 2025).
• Sectional curvature for triples: Through metric comparisons in sampled graphs and point clouds (e.g., Gromov-product and comparison triangle-based radii) (&&&1&&&, Xia, 2021).
• Edge and path curvature in graphs: Using Ricci-type, Forman–Ricci, Haantjes, and Menger curvature, defined algorithmically on edges or small subgraphs (Yadav et al., 26 Oct 2025).
• Curvature on simplicial complexes: Assigning curvatures to faces (triangles), higher-dimensional simplices, or their lower-dimensional boundaries.
• Curvature via information-theoretic or diffusion invariants: Such as entropy defects of heat kernels (&&&4&&&).

Discretization schemes strive for invariance (e.g., under Euclidean or projective transformations), convergence to smooth counterparts, and consistency with topological theorems (Gauss–Bonnet, Chern–Gauss–Bonnet) (Izmestiev, 13 Feb 2025). The mathematical structure of curvature drives both the theoretical properties and algorithmic design of collection procedures.

2. Algorithmic Pipelines for Discrete Surface and Point Cloud Curvature

A canonical pipeline for collecting curvature information on discrete surfaces—exemplified in discrete gravity and simplicial complex geometry (Chamseddine et al., 2024)—proceeds as follows:

1. Surface Sampling and Complex Construction:
   • Embed the surface in ℝ³ and sample according to a fine grid.
   • For each surface point, connect to its four nearest neighbors, forming a local quad.
   • Subdivide each quad into two triangles to produce a pure 2-simplicial complex where each triangle represents a face.
2. Local Metric and Frame Extraction:
   • Compute triangle normals and use them to determine the induced metric components on each face.
   • Construct local zweibeins (orthonormal frames) from the metric.
3. Spin Connection via Torsion-Free Condition:
   • Assign group elements (discrete spin connections) to oriented edges.
   • Impose the discrete torsion-free (Cartan) equations, resulting in coupled nonlinear equations for the spin-connection angles.
4. Discrete Curvature Evaluation:
   • Build holonomies around each triangle (face) to compute the deficit angle.
   • The Gaussian (or sectional) curvature at each triangle is then the deficit angle divided by triangle area.
5. Complexity and Performance:
   • The entire pipeline can be implemented with overall complexity $O(N\log N)$ due to nearest-neighbor search acceleration (e.g., kd-trees).
   • Numerical stability and mesh quality are enhanced by using quasi-uniform, non-degenerate triangles and robust solvers.

This approach enables local curvature assignment at each face while enforcing essential geometric constraints and is validated with convergence to analytic curvature values on standard surfaces (Chamseddine et al., 2024).

3. Curvature Models for Combinatorial Structures and Graphs

Curvature information collection in networks, graphs, and higher-order combinatorial complexes is realized via several families of discrete curvature (Yadav et al., 26 Oct 2025):

• Forman–Ricci curvature: Defined on edges (or higher cells) by combinatorial Bochner–Weitzenböck analogues; for unweighted graphs, $\kappa_F(e) = 4 - \deg(v_1) - \deg(v_2)$ for edge $e=(v_1, v_2)$ (Yadav et al., 26 Oct 2025).
• Ollivier–Ricci curvature: Defined via optimal transport between local probability distributions at endpoints, $\kappa_O(e) = 1 - W_1(m_i, m_j)/d(i, j)$ using the 1-Wasserstein metric (Li et al., 2021, Fu et al., 2024, Yadav et al., 26 Oct 2025).
• Sectional curvature on triples: For a triple $(x_1, x_2, x_3)$, compute Gromov product radii $r_i$, and seek minimal $\rho$ such that the intersection of balls $B(x_i, \rho r_i)$ is nonempty. This $\rho$ quantifies flatness or hyperconvexity and is aggregated into curvature profiles (Beylier et al., 16 Sep 2025, Xia, 2021, Yadav et al., 26 Oct 2025).
• Angle defect and combinatorial curvature: At vertices of planar graphs, combinatorial curvature is given by $$p_C(v) = 1 - \frac{\deg(v)}{2} + \sum_{f\ni v} 1/\deg(f)$$, reflecting angular deficit at $v$ (Izmestiev, 13 Feb 2025, Yadav et al., 26 Oct 2025).
• Mean and Gaussian curvature from point clouds: Estimate tangents/normals via local PCA or Voronoi covariance measures, then assemble shape operators (discrete Weingarten maps) to recover principal, mean, and Gaussian curvatures (Spang, 2023, Mirzaie, 7 Jun 2025).

Efficient linear algebra or local optimization (e.g., solving small least-squares problems or local transport LPs) supports scalable implementation for large graphs or clouds (Spang, 2023, Yadav et al., 26 Oct 2025). Convergence to smooth geometric invariants (with density and regularity assumptions) is established in representative cases (Izmestiev, 13 Feb 2025).

4. Scale-Dependent Profiles and Global Curvature Aggregation

Beyond isolated local values, systematic collection of curvature data across scales or over different configurations is critical for capturing the global geometric characteristics ("curvature fingerprints") of networks and datasets:

• Curvature profiles: Aggregate curvature values, such as the average or histogram of sectional curvature $\rho$ at different sampling scales $r$, to construct a "fingerprint" that characterizes geometry from ultra-metric (tree-like), flat, to positively curved (circular or spherical) domains (Beylier et al., 16 Sep 2025).
• Sampling strategies: For computational tractability, sample equilateral triples, use randomized subsampling, or restrict attention to local neighborhoods. Efficient algorithms incrementally expand balls until intersection occurs, enabling scalable extraction of scale-dependent curvature data (Beylier et al., 16 Sep 2025).
• Visualization and Diagnostics: Scatterplots, boxplots, and Wasserstein distances are used to compare curvature distributions before and after dimensionality reduction, or to estimate intrinsic dimension by locating the embedding dimension minimizing geometric distortion (Beylier et al., 16 Sep 2025).
• Aggregation in geometric learning: Statistics of curvature (mean, min, distribution over support, comparison across graph edges or triangulated faces) serve as robust features for structure-aware unsupervised and supervised learning, manifold recovery, anomaly detection, and model diagnostics (Yadav et al., 26 Oct 2025).

5. Applications in Learning, Optimization, and Physics

Curvature information is essential in a range of contemporary computational and scientific applications:

• Geometric deep learning: Curvature-modulated graph neural networks (e.g., via Ricci curvature-derived edge weights) provide advanced topological sensitivity and improve robustness, message-passing adaptivity, and classification accuracy (Li et al., 2021, Fu et al., 2024, Yadav et al., 26 Oct 2025).
• Network analysis and combinatorics: Community detection, graph rewiring, and structural compression are guided by curvature-driven Ricci flows, combinatorial curvature penalties, and scale-dependent invariants (Yadav et al., 26 Oct 2025).
• Physics and discrete gravity: Simplicial-complex-derived curvature is the basis of Regge-type discretizations in general relativity, network geometry, and causal set theory, enabling numerical studies of discrete spacetime and curvature-induced phenomena (Chamseddine et al., 2024).
• Curvature for optimization: Second-order methods collect and exploit Hessian-based curvature information for accelerated convergence—e.g., curvature-aided gradient tracking (CIAG, A-CIAG) achieves optimal linear and accelerated rates in large-scale convex optimization (Wai et al., 2018, Granziol et al., 2019).
• Curvature in information geometry: Information-theoretic expansions of heat kernels, as in (Sangha, 20 Nov 2025), allow for analytic recovery of scalar and sectional curvature purely from diffusion data, linking geometric invariants to entropy defects.

6. Performance Guarantees, Stability, and Practical Considerations

Modern curvature collection pipelines are characterized by provable rates, error controls, and robustness:

• Convergence and error bounds: Discrete curvature estimators converge to their smooth analogues under mesh regularity, increasing sample density, and suitable algorithmic choices. Explicit sample complexity estimates ensure high-probability existence and accuracy of curvature estimates in point clouds or networks (Mirzaie, 7 Jun 2025).
• Stability to noise and sampling: Robust methods, such as Voronoi covariance smoothing and convolved covariance estimation, demonstrate high resilience to substantial noise and uneven sampling, critical in practical acquisition settings (Spang, 2023).
• Algorithmic complexity: State-of-the-art algorithms scale linearly or near-linearly in the number of points ($O(N \log N)$), or with polynomial dependence for local-neighborhood-based estimators and global linear algebraic solvers (Chamseddine et al., 2024, Yadav et al., 26 Oct 2025).
• Data structure compatibility: Extensive model tables and taxonomies indicate which curvature models apply to which data structures (e.g., graphs, triangulations, simplicial complexes, cubical complexes, point clouds), ensuring broad interoperability (Yadav et al., 26 Oct 2025).
• Limitations and caveats: Extreme nonuniformity or pathological point distributions can degrade discrete curvature estimates; careful mesh quality control, regularization, or pre-processing (e.g., re-sampling) is recommended (Mirzaie, 7 Jun 2025, Spang, 2023).

7. Synthesis and Outlook

Curvature information collection unifies geometric, combinatorial, and information-theoretic perspectives—bridging metric, Riemannian, and discrete geometry. The evolution of rigorous, high-performance algorithms enables systematic measurement and exploitation of curvature in network science, geometric learning, physical modeling, and beyond. As discrete models deepen and computational pipelines diversify, curvature will remain central to geometric signal processing, scalable statistical learning, and fundamental studies of discrete space, topology, and dynamics (Chamseddine et al., 2024, Izmestiev, 13 Feb 2025, Yadav et al., 26 Oct 2025).