论文标题
Sobolev Transport：用于图形指标的概率度量的可扩展度量
Sobolev Transport: A Scalable Metric for Probability Measures with Graph Metrics
论文作者
论文摘要
最佳运输（OT）是比较概率分布的流行措施。但是，OT遭受了一些缺点，例如（i）计算的高复杂性，（ii）不确定，它限制了其对内核机器的适用性。在这项工作中，我们考虑了图公制空间支持的概率度量，并提出了一种新颖的Sobolev运输度量。我们表明，Sobolev转运度量产生了快速计算的封闭式公式，并且是负面的。我们表明，以这种运输距离赋予的概率衡量标准是与具有加权$ \ ell_p $距离的欧几里得空间中设置的有限凸的等距。我们进一步利用Sobolev Transport的负面确定性来设计正定核，并在使用单词嵌入和拓扑数据分析中评估其针对其他基准的性能。
Optimal transport (OT) is a popular measure to compare probability distributions. However, OT suffers a few drawbacks such as (i) a high complexity for computation, (ii) indefiniteness which limits its applicability to kernel machines. In this work, we consider probability measures supported on a graph metric space and propose a novel Sobolev transport metric. We show that the Sobolev transport metric yields a closed-form formula for fast computation and it is negative definite. We show that the space of probability measures endowed with this transport distance is isometric to a bounded convex set in a Euclidean space with a weighted $\ell_p$ distance. We further exploit the negative definiteness of the Sobolev transport to design positive-definite kernels, and evaluate their performances against other baselines in document classification with word embeddings and in topological data analysis.
