Academic

Sobolev--Ricci Curvature

arXiv:2603.12652v1 Announce Type: new Abstract: Ricci curvature is a fundamental concept in differential geometry for encoding local geometric structure, and its graph-based analogues have recently gained prominence as practical tools for reweighting, pruning, and reshaping network geometry. We propose Sobolev-Ricci Curvature (SRC), a graph Ricci curvature canonically induced by Sobolev transport geometry, which admits efficient evaluation via a tree-metric Sobolev structure on neighborhood measures. We establish two consistency behaviors that anchor SRC to classical transport curvature: (i) on trees endowed with the length measure, SRC recovers Ollivier-Ricci curvature (ORC) in the canonical W1 setting, and (ii) SRC vanishes in the Dirac limit, matching the flat case of measure-theoretic Ricci curvature. We demonstrate SRC as a reusable curvature primitive in two representative pipelines. We define Sobolev-Ricci Flow by replacing ORC with SRC in a Ricci-flow-style reweighting rule, a

K
Kyoichi Iwasaki, Tam Le, Hideitsu Hino · · 1 min read · 6 views
#cs.LG

arXiv:2603.12652v1 Announce Type: new Abstract: Ricci curvature is a fundamental concept in differential geometry for encoding local geometric structure, and its graph-based analogues have recently gained prominence as practical tools for reweighting, pruning, and reshaping network geometry. We propose Sobolev-Ricci Curvature (SRC), a graph Ricci curvature canonically induced by Sobolev transport geometry, which admits efficient evaluation via a tree-metric Sobolev structure on neighborhood measures. We establish two consistency behaviors that anchor SRC to classical transport curvature: (i) on trees endowed with the length measure, SRC recovers Ollivier-Ricci curvature (ORC) in the canonical W1 setting, and (ii) SRC vanishes in the Dirac limit, matching the flat case of measure-theoretic Ricci curvature. We demonstrate SRC as a reusable curvature primitive in two representative pipelines. We define Sobolev-Ricci Flow by replacing ORC with SRC in a Ricci-flow-style reweighting rule, and we use SRC for curvature-guided edge pruning aimed at preserving manifold structure. Overall, SRC provides a transport-based foundation for scalable curvature-driven graph transformation and manifold-oriented pruning.

Executive Summary

This article proposes Sobolev-Ricci Curvature (SRC), a graph-based analogue of Ricci curvature, canonically induced by Sobolev transport geometry. SRC is established as a reusable curvature primitive for scalable curvature-driven graph transformation and manifold-oriented pruning. The authors demonstrate SRC in two representative pipelines, including curvature-guided edge pruning and Ricci-flow-style reweighting. SRC is shown to recover Ollivier-Ricci curvature in the canonical W1 setting and to vanish in the Dirac limit, aligning with classical transport curvature. The article presents a novel foundation for graph processing and manifold learning, leveraging the principles of transport geometry.

Key Points

  • ▸ Introduction of Sobolev-Ricci Curvature (SRC), a graph-based analogue of Ricci curvature
  • ▸ Establishment of SRC as a reusable curvature primitive for graph transformation and pruning
  • ▸ Demonstration of SRC in curvature-guided edge pruning and Ricci-flow-style reweighting

Merits

Strength in mathematical foundation

The article builds upon a solid mathematical foundation by leveraging Sobolev transport geometry, ensuring a rigorous and well-reasoned approach to graph processing and manifold learning.

Promising applications in graph processing and manifold learning

The authors demonstrate the potential of SRC in curvature-driven graph transformation and pruning, opening up new avenues for research and development in these areas.

Demerits

Technical complexity and accessibility

The article assumes a high level of mathematical sophistication, which may limit its accessibility to a broader audience, including practitioners in applied fields.

Limited empirical evaluation and validation

The article focuses on theoretical developments and demonstrations, whereas a more comprehensive empirical evaluation and validation of SRC's performance in various scenarios would enhance its practical utility.

Expert Commentary

While the article presents a groundbreaking contribution to the field of differential geometry and its applications in graph processing and manifold learning, it is essential to acknowledge the technical complexity and potential limitations of SRC. The authors' emphasis on mathematical rigor and theoretical developments is commendable, but a more comprehensive empirical evaluation and validation of SRC's performance in various scenarios would be beneficial. Furthermore, collaborations with practitioners in applied fields can help bridge the gap between theoretical developments and practical applications.

Recommendations

  • ✓ Future research should focus on empirical evaluations and validations of SRC's performance in various graph processing and manifold learning scenarios.
  • ✓ Interdisciplinary collaborations between mathematicians, computer scientists, and practitioners in applied fields can help translate the theoretical developments of SRC into practical applications and innovations.

Sources

Related Articles