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This chapter describes the optimal transport problem, with a particular focus on the Wasserstein distances. In the case of the 2- Wasserstein distance, we obtain a fundamental characterization of the structure of the optimal couplings using tools from convex analysis. In particular, we reveal the essential role played by convex duality.

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Optimal Transport

  • 章节

  • 发表日期：

    2025年

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  • Sinho Chewi,
  • Jonathan Niles-Weed,
  • Philippe Rigollet

摘要

This chapter describes the optimal transport problem, with a particular focus on the Wasserstein distances. In the case of the 2- Wasserstein distance, we obtain a fundamental characterization of the structure of the optimal couplings using tools from convex analysis. In particular, we reveal the essential role played by convex duality.