标题： 快速Wasserstein速率用于估计概率图模型的概率分布 显示英文标题

标题： Fast Wasserstein rates for estimating probability distributions of probabilistic graphical models

摘要： 使用独立同分布的数据在Wasserstein距离下估计高维分布是维度灾难的一个基本实例。我们探讨如何利用数据生成过程的结构知识来克服这一困境。更准确地说，我们研究的是已知有向无环图的概率图模型的分布集合。结果表明，这种知识只有在可以量化时才有帮助，我们通过对应于图的分解中的转移核的平滑性条件来形式化这一点。在这种情况下，我们证明了估计速率由图的局部结构决定，更具体地说，由单个节点及其父节点对应的维度决定。精确的速率取决于对核所假设的平滑性的具体概念，其中弱（Wasserstein-Lipschitz）或强（双向总变分-Lipschitz）条件会导致不同的结果。我们在强条件下证明了尖锐性，并展示了该条件对于具有正Lipschitz密度的分布来说是满足的。 显示英文摘要

摘要： Using i.i.d. data to estimate a high-dimensional distribution in Wasserstein distance is a fundamental instance of the curse of dimensionality. We explore how structural knowledge about the data-generating process which gives rise to the distribution can be used to overcome this curse. More precisely, we work with the set of distributions of probabilistic graphical models for a known directed acyclic graph. It turns out that this knowledge is only helpful if it can be quantified, which we formalize via smoothness conditions on the transition kernels in the disintegration corresponding to the graph. In this case, we prove that the rate of estimation is governed by the local structure of the graph, more precisely by dimensions corresponding to single nodes together with their parent nodes. The precise rate depends on the exact notion of smoothness assumed for the kernels, where either weak (Wasserstein-Lipschitz) or strong (bidirectional Total-Variation-Lipschitz) conditions lead to different results. We prove sharpness under the strong condition and show that this condition is satisfied for example for distributions having a positive Lipschitz density.