What does this research mean for the field?

The 2-Wasserstein distance is formally equivalent to the homogeneous Sobolev norm for infinitesimal perturbations, revealing a deep geometric link between optimal transport, harmonic analysis, and Laplacian eigenstructure. Novelty: ClaimNovelty.METHODOLOGICAL. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

To establish the mathematical equivalence between the 2-Wasserstein optimal transport distance and homogeneous negative Sobolev norms for infinitesimal perturbations of probability distributions.

August 30, 2026Open Access

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

View Full Paper

Ask AI

Bookmark

Authors

ACAndrew Stewart Caldin

Discussion

Member takes

Implication

Mathematical analysis demonstrates optimal transport cost equivalence to negative Sobolev norms under infinitesimal perturbations, indicating deep connections to Laplacian harmonic analysis.

Key Points

To establish the mathematical equivalence between the 2-Wasserstein optimal transport distance and homogeneous negative Sobolev norms for infinitesimal perturbations of probability distributions.
Formulated the p-Wasserstein distance using coupling measures over probability distributions on metric spaces.
Analyzed infinitesimal perturbations between measures under the 2-Wasserstein metric relative to density-weighted homogeneous Sobolev spaces.
Demonstrated that the squared 2-Wasserstein distance formally scales as the squared homogeneous negative Sobolev norm of the difference between measures for infinitesimal perturbations.
Identified that Wasserstein geometric structure is governed by the inverse Laplacian, linking transport cost directly to harmonic analysis and Laplacian eigenfunctions.

Cite This Study

Andrew Stewart Caldin (2026) studied this question.

synapsesocial.com/papers/6a93f0ce6c1a8fb52e79d484 https://doi.org/10.5281/zenodo.22138256

View Full Paper

Ask AI

Bookmark

Implication

Mathematical analysis demonstrates optimal transport cost equivalence to negative Sobolev norms under infinitesimal perturbations, indicating deep connections to Laplacian harmonic analysis.

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

Discussion

Member takes

Implication

Key Points

Cite This Study

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

Discussion

Member takes

Implication

Key Points

Cite This Study

Also Consider

Also Consider