Computing Wasserstein barycenters is a fundamental geometric problem with widespread applications in machine learning, statistics, and computer graphics. However, it is unknown whether Wasserstein barycenters can be computed in polynomial time, either exactly or to high precision (i.e., with $\textrm{polylog}(1/\varepsilon)$ runtime dependence). This paper answers these questions in the affirmative for any fixed dimension. Our approach is to solve an exponential-size linear programming formulation by efficiently implementing the corresponding separation oracle using techniques from computational geometry.

中文翻译：

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多项式时间内的高精度 Wasserstein 重心

计算 Wasserstein 重心是一个基本的几何问题，在机器学习、统计学和计算机图形学中有着广泛的应用。然而，Wasserstein 重心是否可以在多项式时间内精确计算或高精度计算（即，具有 $\textrm{polylog}(1/\varepsilon)$ 运行时依赖性）是未知的。本文对任何固定维度都肯定地回答了这些问题。我们的方法是通过使用计算几何技术有效地实现相应的分离预言来解决指数大小的线性规划公式。