Title: Metric Graph Kernels via the Tropical Torelli Map
Primary area: learning on graphs and other geometries & topologies
Keywords: graph kernel, metric graph, tropical geometry, graph 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 spaces. We develop efficient algorithms to compute our kernels and analyze their complexity, which depends primarily on the genus of the input graphs. Empirically, our kernels outperform existing methods in label-free settings, as demonstrated on both synthetic and real-world benchmark datasets. We further showcase their practical utility with an urban road network classification task.

TL;DR: We propose new graph kernels via the tropical Torelli map, comparing graphs through intrinsic geometry and topological cycles instead of nodes or edges.