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In the previous two chapters, we studied the space \(\mathcal {W}_{2} = (\mathcal {P}_{2}(\mathbb {R}^{d}), W_{2})\) through the lens of Riemannian geometry. Although such an approach yields considerable geometric insight and can even be treated rigorously (see [18]), it is important to keep in mind that \(\mathcal {W}_2\) is not a bona fide Riemannian manifold, and consequently technical issues abound. In this chapter, we seek to study the metric geometry of Wasserstein space directly.

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Metric Geometry of the Wasserstein Space

  • 章节

  • 发表日期：

    2025年

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

摘要

In the previous two chapters, we studied the space \(\mathcal {W}_{2} = (\mathcal {P}_{2}(\mathbb {R}^{d}), W_{2})\) through the lens of Riemannian geometry. Although such an approach yields considerable geometric insight and can even be treated rigorously (see [18]), it is important to keep in mind that \(\mathcal {W}_2\) is not a bona fide Riemannian manifold, and consequently technical issues abound. In this chapter, we seek to study the metric geometry of Wasserstein space directly.