Wasserstein Metric as Linearized Optimal Transport Sobolev Norm — E8 Intelligence Research

Abstract

FINDING: The Wasserstein metric (Earth Mover's Distance) provides an optimal-transport geometry on probability distributions, formally equivalent to a weighted homogeneous Sobolev norm for infinitesimal perturbations. | MATH: For probability measures μ,ν on metric space (X,d), the p-Wasserstein distance is W_p(μ,ν) = (inf_{γ∈Π(μ,ν)} ∫ d(x,y)^p dγ(x,y))^{1/p}. For p=2, W_2²(μ,ν) ≈ ‖μ−ν‖²_{Ḣ^{-1}} (weighted homogeneous Sobolev norm) for small perturbations — the linearized optimal transport metric. The dual Kantorovich formulation: W_1(μ,ν) = sup_{‖f‖_Lip≤1} (∫ f dμ − ∫ f dν). | CONNECTION: The W_2 metric induces a Riemannian structure on the space of densities (Otto calculus), with geodesics corresponding to gradient flows of entropy. This is the Wasserstein-2 geometry underlying the continuity equation ∂_t ρ + ∇·(ρ∇φ)=0. The linearization to Ḣ^{-1} reveals a natural inner product structure — a Hilbert space geometry that is the infinite-dimensional analogue of Euclidean space. No direc Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com