Tropical Geometry Enhances Neuronal Graph Learning Expressivity

A new training-free geometric prior, rooted in tropical algebraic geometry, has been developed to enhance the representation of 3D neuronal morphologies for graph learning. Current Graph Neural Networks (GNNs) often struggle to capture intricate topological features like cycles induced by spatial proximities, being limited by the 1-Weisfeiler-Lehman test. This research addresses this by applying tropical Abel-Jacobi transforms and polarization distances to machine learning on tree-structured data. The core of the approach involves a structural transformation pipeline that converts spatial trees into cyclic metric graphs, making them suitable for embedding into the Tropical Jacobian. To overcome the computational challenge of solving the NP-Hard Closest Vector Problem for exact tropical polarization distances, the method uses a continuous relaxation on the universal cover of the Albanese torus. This yields a discrete Arakelov-Green measure, which decomposes into an intrinsic path metric and an unquantized polarization distance. The resulting descriptors—node-level structural coordinates from eigenvectors and a graph-level signature from the eigenvalue spectrum—demonstrate expressivity beyond the 1-WL limit on benchmarks and significantly improve classification accuracy on 3D morphology datasets when integrated into standard architectures.