Computing the Tropical Abel--Jacobi Transform and Tropical Distances for Metric Graphs

TL;DR

This work introduces a computational pipeline for metric graphs treated as abstract tropical curves, enabling a vectorization via the tropical Abel–Jacobi transform into the tropical Jacobian. It develops concrete algorithms based on cycle–edge and path–edge incidence matrices, with interpolation to sample points on the transform and analyzes how different combinatorial models affect embeddings. The paper defines two distance notions on the tropical Jacobian, proves NP-hardness for general distance computation, and offers exact solutions in special cases alongside practical approximations using lattice reduction and MILP, complemented by simulations and public code. Together, these results establish a foundation for applying tropical geometry to machine learning and data analysis on metric-graph-structured data.

Abstract

Metric graphs are important models for capturing the structure of complex data across various domains. While much effort has been devoted to extracting geometric and topological features from graph data, computational aspects of metric graphs as abstract tropical curves remains unexplored. In this paper, we present the first computational and machine learning-driven study of metric graphs from the perspective of tropical algebraic geometry. Specifically, we study the tropical Abel--Jacobi transform, a vectorization of points on a metric graph via the tropical Abel--Jacobi map into its associated flat torus, the tropical Jacobian. We develop algorithms to compute this transform and investigate how the resulting embeddings depend on different combinatorial models of the same metric graph. Once embedded, we compute pairwise distances between points in the tropical Jacobian under two natural metrics: the tropical polarization distance and the Foster--Zhang distance. Computing these distances are generally NP-hard as they turn out to be linked to classical lattice problems in computational complexity, however, we identify a class of metric graphs where fast and explicit computations are feasible. For the general case, we propose practical algorithms for both exact and approximate distance matrix computations using lattice basis reduction and mixed-integer programming solvers. Our work lays the groundwork for future applications of tropical geometry and the tropical Abel--Jacobi transform in machine learning and data analysis.

Figures (10)

• Figure 1: An illustrative example of our proposed framework. (a) We begin with a combinatorial model $G$ of a random metric graph $\Gamma$. (b) We compute the tropical Abel--Jacobi transform of $G$ and sample additional points via interpolation. The dotted lines represent the fundamental domain of the tropical Jacobian and the blue points correspond to the tropical Abel--Jacobi transform of $\Gamma$. (c) We compute the pairwise distance matrix for the point cloud data sampled from the tropical Jacobian. The distance matrix is visualized via multidimensional scaling (MDS).
• Figure 2: A tropical atlas corresponding to the length metric on $\Gamma$. The metric graph is constructed by joining three unit intervals at two end points $p$ and $q$. Red lines indicate the open neighborhood $U_p=\Gamma\backslash \{q\}$ and the primitive lattice vectors under the embedding $\phi_{U_p}$. Blue lines indicate the open neighborhood $U_q=\Gamma\backslash\{p\}$ and primitive lattice vectors under the embedding $\phi_{U_q}$.
• Figure 3: A metric graph and its tropical Abel--Jacobi transform. On the left panel, a metric graph is represented by a combinatorial model with 3 vertices and 5 edges. The fundamental 1-cycles $\sigma_1,\sigma_2,\sigma_3$ are drawn in red dashed lines. On the right panel, the tropical Abel--Jacobi transform of the metric graph is the piecewise linear curve colored in blue. The $\tau_i$'s are lattice vectors corresponding to the $\sigma_i$'s. A fundamental domain of the tropical Jacobian is shaded in green. For the purpose of visualization, all $v_i$'s and $u_i$'s are translated by $[1,0,1]^\top$, so that they reside in the first quadrant and the fundamental domain.
• Figure 4: Illustration of Simplification. On the left panel, a combinatorial model with 8 vertices and 10 edges is shown. On the right panel, after contracting bridge edges and deleting vertices of valence 2, the simplified combinatorial model has only 3 vertices and 5 edges.
• Figure 5: Computation time for tropical polarization distance matrices. The left/right panel shows a log-log plot of computation time versus the number of nodes/graph genus. Different colors represent algorithms for solving the exact CVP.

Theorems & Definitions (71)

• Remark 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Remark 2.6
• Theorem 2.7
• proof
• Example 2.8
• Definition 3.1