Wasserstein Metric as Optimal Transport with Sobolev Link — E8 Intelligence Research

FINDING: The Wasserstein metric (Earth Mover's Distance) provides a geometrically grounded distance between probability distributions, equivalent to optimal transport cost, with a formal link to Sobolev norms for infinitesimal perturbations. | MATH: For probability measures μ, ν on metric space (X,d), the p-Wasserstein distance is \( W_p(\mu,\nu) = \left( \inf_{\gamma \in \Gamma(\mu,\nu)} \int_{X \times X} d(x,y)^p \, d\gamma(x,y) \right)^{1/p} \). For p=2, \( W_2^2(\mu,\nu) \) is formally equivalent to the homogeneous Sobolev norm \( \dot{H}^{-1} \) for infinitesimal perturbations: \( W_2^2(\mu,\nu) \sim \| \mu - \nu \|_{\dot{H}^{-1}}^2 \) (weighted by density). The optimal transport plan γ is a coupling; the cost is the minimal "work" = mass × distance. | CONNECTION: The \( \dot{H}^{-1} \) equivalence reveals a deep link to harmonic analysis and Laplacian eigenstructure — the Wasserstein geometry is governed by the inverse Laplacian, which in Euclidean space has eigenfunctions with c Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com