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

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Authors

Publication Details

Journal
Zenodo (CERN European Organization for Nuclear Research)
Published
2026-08-28
DOI
https://doi.org/10.5281/zenodo.22138257
Primary Topic
Geometric Analysis and Curvature Flows
Type
preprint
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preprint

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

Andrew Stewart Caldin
Zenodo (CERN European Organization for Nuclear Research)
Geometric Analysis and Curvature Flows
preprint

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

Andrew Stewart Caldin
preprint en

Abstract

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

Zenodo (CERN European Organization for Nuclear Research)
Geometric Analysis and Curvature Flows
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Wasserstein Metric as Optimal Transport with Sobolev Link — E8 Intelligence Research — Andrew Stewart Caldin · Zenodo (CERN European Organization for Nuclear Research) (2026) | TGRS Research Map | TGRS