Computing the Tropical Abel--Jacobi Transform and Tropical Distances for Metric Graphs (2504.11619v2)

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.

Tropical Abel–Jacobi Transform and Tropical Distances for Metric Graphs

In the context of tropical algebraic geometry, this paper focuses on computational aspects of metric graphs, exploring them as abstract tropical curves through the lens of the tropical Abel–Jacobi transform. This is a novel computational and machine learning-oriented paper devoted to expanding the potential applications of tropical geometry in data science.

Metric Graphs as Abstract Tropical Curves

Metric graphs, key components in modeling complex data, are structures combining the combinatorial nature of graphs with the geometric framework of metric spaces. These entities are prevalent across numerous disciplines, from pure mathematics to machine learning and biology. However, their computational exploration as tropical curves is yet to be fully realized in practice. This paper pivots on this gap, offering new methods to vectorize such graphs via the tropical Abel–Jacobi map—a technique that represents points on a metric graph within its associated tropical Jacobian, a flat torus.

Computational Framework and Algorithms

The research introduces algorithms to compute the tropical Abel–Jacobi transform of metric graphs. This transform is crucial as it offers a systematic way to create vector representations of graphs in the tropical Jacobian, potentially enhancing the efficiency of data analysis tasks by embedding points into an algebraic structure known for its rich properties.

1. Cycle-Edge and Path-Edge Incidence Matrices: The paper introduces the use of cycle-edge and path-edge incidence matrices to represent the relationships in a combinatorial graph. These matrices form the backbone for computing the tropical Abel–Jacobi transform.
2. Distance Measures: Two natural metrics—the tropical polarization distance and the Foster–Zhang distance—are explored for embedding points. However, computing these distances is typically NP-hard, tied to classical lattice problems in computational complexity.
3. Optimization Solutions: Practical solutions via lattice basis reduction and mixed-integer programming are presented, especially targeting graphs where quick computations are feasible. The algorithms allow for both exact and approximate calculations, making them versatile in application.

Theoretical and Practical Implications

The paper broadens the theoretical understanding of metric graphs as tropical curves. Insights into altering combinatorial models, detecting graph properties (such as identifying bridges) via cycle-edge matrices, and improving computational efficiency through structural simplifications are extensively discussed. These computational advancements are pivotal for the practical application of tropical geometry in machine learning and data science.

Future Directions and Applications

Anticipating further developments, the paper hints at significant potential extensions such as:

• Topological Data Analysis (TDA): With metric graphs playing critical roles in TDA, the tropical transformation may enhance the extraction of persistent homology features, streamlining computational complexity.
• Probabilistic Models and Random Graphs: The tropical Abel–Jacobi framework can be extended to paper random graph models or statistical behaviors of complex networks through probabilistic subsampling.
• Generalization Beyond Graphs: The methods open avenues for generalizing such studies to higher-dimensional polyhedral spaces and even more abstract structures in tropical geometry, which could have implications in statistical topology and machine learning on complex data shapes.

By exploring the computational facets of metric graphs framed as tropical objects, this research merges the abstract mathematical constructs with applicable data-driven algorithms, setting a groundwork for further exploration and integration of tropical mathematics with computational sciences.