论文标题
统计最佳运输作为学习内核嵌入
Statistical Optimal Transport posed as Learning Kernel Embedding
论文作者
论文摘要
统计最佳运输（OT）的目的是仅使用给定源和目标边缘分布的样品始终估算最佳运输计划/地图。这项工作采用了提出统计OT的新方法，因为学习运输计划的内核平均值是从基于样本的边缘嵌入估计值中嵌入的。提出的估计器控制通过采用最大平均基于基于平均差异的正规化，这与$ ϕ $ -DIVERGENCE（熵）的互补互补，该正则化通常在现有估计器中使用。一个关键结果是，在非常温和的条件下，运输计划的$ε$最佳恢复以及基于Barycentric的传输图是可以完全不含尺寸的样品复杂性的。此外，内核平均嵌入中的隐式平滑可以实现样本外估计。事实证明，适当的代表定理导致估算器的内核凸公式，然后即使在非标准域中也可以使用该公式。经验结果说明了所提出的方法的功效。
The objective in statistical Optimal Transport (OT) is to consistently estimate the optimal transport plan/map solely using samples from the given source and target marginal distributions. This work takes the novel approach of posing statistical OT as that of learning the transport plan's kernel mean embedding from sample based estimates of marginal embeddings. The proposed estimator controls overfitting by employing maximum mean discrepancy based regularization, which is complementary to $ϕ$-divergence (entropy) based regularization popularly employed in existing estimators. A key result is that, under very mild conditions, $ε$-optimal recovery of the transport plan as well as the Barycentric-projection based transport map is possible with a sample complexity that is completely dimension-free. Moreover, the implicit smoothing in the kernel mean embeddings enables out-of-sample estimation. An appropriate representer theorem is proved leading to a kernelized convex formulation for the estimator, which can then be potentially used to perform OT even in non-standard domains. Empirical results illustrate the efficacy of the proposed approach.
