论文标题
高斯平滑的最佳运输：度量结构和统计效率
Gaussian-Smooth Optimal Transport: Metric Structure and Statistical Efficiency
论文作者
论文摘要
最佳运输（OT），尤其是Wasserstein距离，已经看到了机器学习中的兴趣和应用。然而，瓦瑟尔斯坦距离下的经验近似受到严重的维度诅咒，使它们在高维度上不切实际。结果，熵正规的OT已成为流行的解决方法。但是，尽管它具有快速的算法和更好的统计属性，但它却散发出了瓦斯坦斯坦距离所享受的公制结构。这项工作提出了一种新颖的高斯木ot（GoT）框架，它实现了两者中最好的：保留1-wasserstein度量结构，同时减轻了维度的经验近似诅咒。此外，随着高斯平滑参数的收缩缩小到零，获得了$γ$ - 对经典ot的融合（具有优化器的收敛），从而作为天然扩展。提供了一项支持理论结果的经验研究，促进了高斯平滑的OT作为熵OT的强大替代方案。
Optimal transport (OT), and in particular the Wasserstein distance, has seen a surge of interest and applications in machine learning. However, empirical approximation under Wasserstein distances suffers from a severe curse of dimensionality, rendering them impractical in high dimensions. As a result, entropically regularized OT has become a popular workaround. However, while it enjoys fast algorithms and better statistical properties, it looses the metric structure that Wasserstein distances enjoy. This work proposes a novel Gaussian-smoothed OT (GOT) framework, that achieves the best of both worlds: preserving the 1-Wasserstein metric structure while alleviating the empirical approximation curse of dimensionality. Furthermore, as the Gaussian-smoothing parameter shrinks to zero, GOT $Γ$-converges towards classic OT (with convergence of optimizers), thus serving as a natural extension. An empirical study that supports the theoretical results is provided, promoting Gaussian-smoothed OT as a powerful alternative to entropic OT.
