What does this research mean for the field?

The Wasserstein metric provides an optimal-transport geometry on probability distributions that is formally equivalent to a weighted homogeneous Sobolev norm for infinitesimal perturbations. Novelty: ClaimNovelty.METHODOLOGICAL. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

To establish the mathematical correspondence between the quadratic Wasserstein metric and weighted Sobolev norms under infinitesimal density perturbations.

August 30, 2026Open Access

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

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Authors

ACAndrew Stewart Caldin

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Implication

Mathematical analysis demonstrates an equivalence between Wasserstein metrics and Sobolev norms in probability distributions, indicating a flat Hilbert space geometry in optimal transport.

Key Points

To establish the mathematical correspondence between the quadratic Wasserstein metric and weighted Sobolev norms under infinitesimal density perturbations.
Evaluated the 2-Wasserstein metric and its dual Kantorovich formulation for probability measures on metric spaces.
Linearized the optimal transport geometry around reference densities using Otto calculus and the continuity equation.
Demonstrated that the squared 2-Wasserstein distance formally approximates a weighted homogeneous Sobolev norm for small perturbations.
Showed that the Wasserstein-2 metric induces a Riemannian manifold on density spaces governed by gradient flows of entropy.
Revealed an inner product structure in the linearized regime that equates optimal transport perturbations to an infinite-dimensional Hilbert space.

Cite This Study

Andrew Stewart Caldin (2026) studied this question.

synapsesocial.com/papers/6a941ce46c1a8fb52e79e754 https://doi.org/10.5281/zenodo.22154690

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Implication

Mathematical analysis demonstrates an equivalence between Wasserstein metrics and Sobolev norms in probability distributions, indicating a flat Hilbert space geometry in optimal transport.

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

Discussion

Member takes

Implication

Key Points

Cite This Study

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

Discussion

Member takes

Implication

Key Points

Cite This Study

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