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?

Characterize the geometric relationship between the 2-Wasserstein optimal transport metric and weighted homogeneous Sobolev norms under infinitesimal perturbations.

August 30, 2026Open Access

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

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Characterize the geometric relationship between the 2-Wasserstein optimal transport metric and weighted homogeneous Sobolev norms under infinitesimal perturbations.
Evaluated the Kantorovich dual formulation and quadratic 2-Wasserstein distance across probability measures on metric spaces.
Applied Otto calculus to linearize optimal transport dynamics via the continuity equation.
The squared 2-Wasserstein distance approximates the weighted homogeneous Sobolev norm for small perturbations, establishing the linearized optimal transport metric.
The 2-Wasserstein metric induces a Riemannian structure on probability density spaces, aligning geodesics with entropy gradient flows.
Linearization to the negative-one homogeneous Sobolev space reveals an inner-product Hilbert space structure as an infinite-dimensional Euclidean analogue.

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Implication

Theoretical analysis demonstrates the equivalence of the Wasserstein-2 metric and weighted Sobolev norms for infinitesimal perturbations, highlighting a Hilbert space geometry in optimal transport.

Cite This Study

Andrew Stewart Caldin (2026) studied this question.

synapsesocial.com/papers/6a93f1176c1a8fb52e79dc4b https://doi.org/10.5281/zenodo.22154691

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Implication

Theoretical analysis demonstrates the equivalence of the Wasserstein-2 metric and weighted Sobolev norms for infinitesimal perturbations, highlighting a Hilbert space geometry in optimal transport.

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

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Discussion

Implication

Cite This Study

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

Key Points

Discussion

Implication

Cite This Study

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