Bibliographic Information: Emami, P., & Pass, B. (2024). Optimal transport with optimal transport cost: The Monge–Kantorovich problem on Wasserstein spaces. arXiv preprint arXiv:2406.08585v2.
Research Objective: This paper investigates the Monge-Kantorovich optimal transport problem on Wasserstein spaces, aiming to establish the existence and uniqueness of optimal transport maps when the cost function is the squared Wasserstein distance.
Methodology: The authors leverage a recent result by Dello Schiavo (2020), which provides a Rademacher-type theorem for Wasserstein spaces. This theorem, alongside classical optimal transport theory and properties of Riemannian manifolds, forms the basis for their proof.
Key Findings: The paper demonstrates that under the assumption of absolute continuity of the source measure with respect to a reference measure satisfying the Rademacher property, a unique optimal transport plan exists. This plan is further shown to be induced by an optimal map. The authors extend this result to a broader class of cost functions, specifically those expressible as strictly increasing and strictly convex functions of the squared Wasserstein distance.
Main Conclusions: The study significantly contributes to the understanding of optimal transport in the context of Wasserstein spaces. By proving the existence and uniqueness of optimal maps for a relevant class of cost functions, it paves the way for further research and applications in areas where the objects of interest are probability distributions themselves.
Significance: This work has implications for fields dealing with uncertainty and ambiguity in data, such as machine learning, statistics, and economics. It provides a theoretical framework for analyzing and comparing probability distributions using the Wasserstein metric as a cost function.
Limitations and Future Research: The results rely on specific assumptions regarding the reference measure and the source measure. Future research could explore relaxing these assumptions or investigating the problem for different cost functions and underlying spaces. Additionally, exploring practical applications of these theoretical findings in fields like machine learning and data analysis would be valuable.
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by Pedram Emami... às arxiv.org 10-10-2024
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