Tree-Sliced Variants of Wasserstein Distances
Tree-Sliced Variants of Wasserstein Distances
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DOI：
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发表时间：
2019-02
影响因子：
5.4
通讯作者：
Tam Le;M. Yamada;K. Fukumizu;Marco Cuturi
中科院分区：
文献类型：
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作者：
Tam Le;M. Yamada;K. Fukumizu;Marco Cuturi
最优运输（OT）理论定义了一套强大的工具来比较概率分布。然而，OT~遭受一些缺点，计算和统计，这鼓励了OT在最近的文献中的几个正则化变体的建议，其中最值得注意的是\textit{切片}公式，它利用了一元分布之间的封闭形式公式，通过将高维测量投影到随机线上。在这项工作中，我们考虑一个更一般的地面度量家族，即\textit{tree metrics}，它也可以产生快速的闭合形式计算和负定，其中切片沃瑟斯坦距离是一个特例（树是一个链）。我们提出了树切片Wasserstein距离，通过平均这些措施之间的Wasserstein距离使用随机树度量，自适应地建立在低维或高维空间计算。利用该距离的负定性，我们还提出了一个正定核，并在一些基准任务上对其他基线进行了测试。
Optimal transport (\OT) theory defines a powerful set of tools to compare probability distributions. \OT~suffers however from a few drawbacks, computational and statistical, which have encouraged the proposal of several regularized variants of OT in the recent literature, one of the most notable being the \textit{sliced} formulation, which exploits the closed-form formula between univariate distributions by projecting high-dimensional measures onto random lines. We consider in this work a more general family of ground metrics, namely \textit{tree metrics}, which also yield fast closed-form computations and negative definite, and of which the sliced-Wasserstein distance is a particular case (the tree is a chain). We propose the tree-sliced Wasserstein distance, computed by averaging the Wasserstein distance between these measures using random tree metrics, built adaptively in either low or high-dimensional spaces. Exploiting the negative definiteness of that distance, we also propose a positive definite kernel, and test it against other baselines on a few benchmark tasks.