CLEARMIND: Closed-Form Laplacian Embeddings of Arakelov-Green Resistance for Morphological Intrinsic Neuronal Distances
Tropical geometry turns a metric graph into a flat torus, its tropical Jacobian, and measures distances between the images of its points by a closest vector problem on the period lattice. We prove that for points of the graph this problem is solved by geodesics: the squared tropical polarization distance equals the path metric minus the Arakelov-Green (AG) distance, an effective-resistance kernel given in closed form by the Laplacian pseudoinverse. Equivalently, the AG distance and the path metric are the minimal energies of real and of integral unit flows, and the squared tropical distance is exactly their integrality gap. The full tropical distance matrix is thus computable in cubic tim...
| Authors | Yuyang Zhang, Weihan Xu, Xuehai Zhou et al. |
|---|---|
| Primary category | cs.LG |
| Published | 5 August 2026 |
| Last updated at source | 3 October 2026 |
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