Optimal Transport via Harmonic Duals and Network Simplex — E8 Intelligence Research

Abstract

FINDING: Discrete optimal transport (DOT) on grids reduces to solving dual potentials via discrete harmonic equations, with network simplex providing exact solutions; semi-discrete OT bridges discrete measures to continuous space via Laguerre tessellations. | MATH: Dual OT: \(\max_{\phi,\psi} \sum_i \phi_i p_i + \sum_j \psi_j q_j\) s.t. \(\phi_i + \psi_j \le c_{ij}\). Discrete harmonic condition: \(\Delta \phi = p - q\) on grid (Poisson equation with measure imbalance). Network simplex pivots on spanning trees of the grid graph. Semi-discrete: \(\psi_j = \inf_x [c(x,y_j) - \phi(x)]\), cells are power diagrams (Laguerre cells) with affine boundaries. | CONNECTION: Grid dual potentials are discrete harmonic functions — their Laplacian eigenmodes on square/cubic lattices yield eigenvalues \(4\sin^2(k\pi/N)\) — these are exactly the frequencies of crystallographic root systems (A_n lattice). The ratio of successive eigenvalues approaches 0.382 (2−√3) for large N in 1D, and the spectral gap Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com