The Sinkhorn algorithm, parabolic optimal transport and geometric Monge–Ampère equations
The Sinkhorn algorithm, parabolic optimal transport and geometric Monge–Ampère equations
复制标题
Sinkhorn 算法、抛物线最优传输和几何 Monge-Ampère 方程
DOI：
10.1007/s00211-020-01127-x
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发表时间：
2017
影响因子：
2.1
通讯作者：
R. Berman
中科院分区：
文献类型：
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作者：
R. Berman
我们表明，离散Sinkhorn算法的设置中应用的最佳运输的紧凑的流形收敛到一个完全非线性抛物型偏微分方程的解决方案的蒙格-安培型，在一个大规模的限制。后一个演化方程以前出现在不同的背景下（例如，在环面上，它可以与里奇流相一致）。这导致算法近似的潜力的最佳运输地图，以及最佳运输距离，明确的界限上的算术复杂性的建设和近似误差。作为应用程序，我们获得了显式计划的近线性复杂性，在每次迭代中，最佳运输的环面和两个球体，以及远场天线问题。利用准蒙特卡罗方法的连接。
We show that the discrete Sinkhorn algorithm—as applied in the setting of Optimal Transport on a compact manifold—converges to the solution of a fully non-linear parabolic PDE of Monge–Ampère type, in a large-scale limit. The latter evolution equation has previously appeared in different contexts (e.g. on the torus it can be be identified with the Ricci flow). This leads to algorithmic approximations of the potential of the Optimal Transport map, as well as the Optimal Transport distance, with explicit bounds on the arithmetic complexity of the construction and the approximation errors. As applications we obtain explicit schemes of nearly linear complexity, at each iteration, for optimal transport on the torus and the two-sphere, as well as the far-field antenna problem. Connections to Quasi-Monte Carlo methods are exploited.
DOI：
10.1016/j.jcp.2015.12.018
发表时间：
2015-12
期刊：
J. Comput. Phys.
影响因子：
--
作者：
H. Weller;P. Browne;C. Budd;M. Cullen
通讯作者：
H. Weller;P. Browne;C. Budd;M. Cullen