What does this research mean for the field?

Optimal transport (Wasserstein distance) provides a metric structure on probability spaces, with Kantorovich-Rubinstein duality linking it to Lipschitz functions, forming a fundamental bridge between geometry and information. Novelty: ClaimNovelty.SYNTHESIS. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

Characterize the metric and geometric framework of optimal transport and evaluate how Wasserstein distances interface with Lipschitz duality, Riemannian geometry, and information metrics.

August 30, 2026Open Access

Wasserstein Geometry: Bridging Probability and Lipschitz Duality — E8 Intelligence Research

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Characterize the metric and geometric framework of optimal transport and evaluate how Wasserstein distances interface with Lipschitz duality, Riemannian geometry, and information metrics.
Formulated the Wasserstein-p distance and applied Kantorovich-Rubinstein duality for p=1 across probability measures.
Analyzed the quadratic Wasserstein space (W2) as an Alexandrov geodesic space with non-negative curvature using Otto calculus.
Demonstrated that Kantorovich-Rubinstein duality equates W1 distance to the supremum of differences across 1-Lipschitz functions.
Showed that the squared W2 distance induces a Riemannian-like structure with an L2 tangent space metric tensor g_rho = div(rho grad).
Connected the Otto calculus framework to the Fisher-Rao information metric and the spectral gap of the base space Laplacian.

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Implication

Theoretical analysis reveals metric structures linking probability measures to Lipschitz functions in probability spaces, highlighting deep bridges between optimal transport and information theory.

Cite This Study

Andrew Stewart Caldin (2026) studied this question.

synapsesocial.com/papers/6a93f0f56c1a8fb52e79d84b https://doi.org/10.5281/zenodo.22138480

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Implication

Theoretical analysis reveals metric structures linking probability measures to Lipschitz functions in probability spaces, highlighting deep bridges between optimal transport and information theory.

Wasserstein Geometry: Bridging Probability and Lipschitz Duality — E8 Intelligence Research

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Discussion

Implication

Cite This Study

Wasserstein Geometry: Bridging Probability and Lipschitz Duality — E8 Intelligence Research

Key Points

Discussion

Implication

Cite This Study

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