SIAM Journal on Optimization, Volume 30, Issue 3, Page 2441-2469, January 2020.
In this work, we present a method to compute the Kantorovich--Wasserstein distance of order 1 between a pair of two-dimensional histograms. Recent works in computer vision and machine learning have shown the benefits of measuring Wasserstein distances of order 1 between histograms with $n$ bins by solving a classical transportation problem on very large complete bipartite graphs with $n$ nodes and $n^2$ edges. The main contribution of our work is to approximate the original transportation problem by an uncapacitated min cost flow problem on a reduced flow network of size $O(n)$ that exploits the geometric structure of the cost function. More precisely, when the distance among the bin centers is measured with the 1-norm or the $\infty$-norm, our approach provides an optimal solution. When the distance among bins is measured with the 2-norm, (i) we derive a quantitative estimate on the error between optimal and approximate solution; (ii) given the error, we construct a reduced flow network of size $O(n)$.

中文翻译：

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用无容量最小成本流计算二维直方图之间的Kantorovich-Wasserstein距离

SIAM优化杂志，第30卷，第3期，第2441-2469页，2020年1月。
在这项工作中，我们提出了一种计算一对二维直方图之间的阶1的Kantorovich-Wasserstein距离的方法。最近在计算机视觉和机器学习中的工作表明，通过在带有$ n $节点和$ n ^ 2 $边的非常大的完整二部图上求解经典的运输问题，可以测量带有$ n $箱的直方图之间的Wasserstein距离1的好处。 。我们工作的主要贡献是在大小为$ O（n）$的缩减流量网络上，通过一个无能力的最小成本流问题来近似原始运输问题，该网络利用了成本函数的几何结构。更准确地说，当用1范数或$ \ infty $范数测量箱中心之间的距离时，我们的方法提供了最佳解决方案。当使用2范数来衡量仓之间的距离时，（i）对最优解和近似解之间的误差进行定量估计；（ii）考虑到错误，我们构造了一个大小为$ O（n）$的简化流量网络。