Metric Graph Kernels via the Tropical Torelli Map
Yueqi Cao, Anthea Monod

TL;DR
This paper introduces novel metric graph kernels derived from tropical algebraic geometry, enabling geometry-based comparison of metric graphs that are invariant under edge subdivision, with demonstrated effectiveness on synthetic and real-world data.
Contribution
It presents the first graph kernels based on tropical geometry for metric graphs, offering invariance under edge subdivision and efficient algorithms for computation.
Findings
Kernels capture geometric and topological features overlooked by combinatorial methods.
Algorithms' complexity depends mainly on graph genus, not size.
Effective in real-world urban network classification.
Abstract
We introduce the first graph kernels for metric graphs via tropical algebraic geometry. In contrast to conventional graph kernels based on graph combinatorics such as nodes, edges, and subgraphs, our metric graph kernels are purely based on the geometry and topology of the underlying metric space. A key characterizing property of our construction is its invariance under edge subdivision, making the kernels intrinsically well-suited for comparing graphs representing different underlying metric spaces. We develop efficient algorithms to compute our kernels and analyze their complexity, which depends primarily on the genus of the input graphs rather than their size. Through experiments on synthetic data and selected real-world datasets, we demonstrate that our kernels capture complementary geometric and topological information overseen by standard combinatorial approaches, particularly in…
Peer Reviews
Decision·Submitted to ICLR 2026
* Novelty and rigor: Introduces a kernel framework grounded in tropical geometry. Bridges algebraic topology (graph homology) and SPD information geometry. The paper rigorously shows that the tropical Torelli map yields a unique and refinement-invariant SPD representation for graphs with generic edge lengths. * Empirical performance: Across 23 benchmark datasets, TTE and TTW match or slightly outperform existing unlabeled kernels (typically by 2–8 pp). Results on urban road network (URN) classif
* Limited theoretical depth as a kernel paper: Beyond the refinement-invariance theorem, the work does not establish standard kernel properties (e.g., conditional positive definiteness, universality, or injectivity). The kernels are defined rather than theoretically characterized. * Modest empirical gains: The improvements over strong label-free baselines (Graphlet, Shortest Path, k-Core) are consistent but small. Results are competitive rather than outstanding. * Scalability constraints: The m
- The idea of linking tropical geometry and graph kernels is original and well-grounded mathematically. - Theoretical properties, including invariance under refinements, are clearly presented and proven. - The algorithmic formulation is concrete and supported by detailed complexity analysis. - Experiments are broad and carefully executed, with strong performance in label-free settings.
- There is no comparison with the Weisfeiler–Lehman (WL) kernel, which is the main baseline in graph kernel research. For example, Wasserstein WL (WWL) (Togninalli et al., NeurIPS 2019) would make the results more complete. - Runtime grows fast with graph density; scalability to large or dense graphs remains unclear. Togninalli M., Ghisu E., Llinares-López F., Rieck B., and Borgwardt K. M. (2019). Wasserstein Weisfeiler–Lehman graph kernels. NeurIPS 2019.
A key strength is the new usage of the Tropical Torelli Map to create metric graph kernels, which are fundamentally different from conventional methods because they are purely based on the geometry and topology of the underlying metric space. This construction results in kernels that are invariant under edge subdivision. This new geometric approach establishes the first framework specifically designed to compare and classify metric graphs based on their intrinsic structure.
The two main weaknesses I see are 1. concern the novelty of the underlying mathematical concepts and 2. the availability of robust, large-scale experiments. While the application to graph kernels for metric graphs is new, the core components are built on established fields: the embedding targets a space of SPD (Symmetric Positive Definite) matrices, which is a heavily studied manifold in information geometry. Similarly, the distance function used for the Tropical Torelli-Wasserstein (TTW) kernel
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Taxonomy
TopicsAdvanced Graph Neural Networks · Graph Theory and Algorithms · Graph Labeling and Dimension Problems
