We present an iterative method to efficiently solve the optimal
transportation problem for a class of strictly convex costs which includes
quadratic and p-power costs. Given two probability measures supported on a
discrete grid with n points, we compute the optimal map using O(n) storage
space and O(n log(n)) operations per iteration, with an approximately
exponential convergence rate. Our approach allows us to solve optimal
transportation problems on spatial grids as large as 4096x4096 and 384x384x384
in a matter of minutes.

本研究提出一种迭代方法来高效地解决一类严格凸代价函数的最优运输问题，该方法包括二次和 p 次幂代价函数。我们针对两个定义在离散网格上的概率分布，使用 O (n) 的存储空间和 O (n log (n)) 的操作量计算最优映射，具有近似指数收敛速度。该方法能够在几分钟内解决空间网格大小达到 4096x4096 和 384x384x384 的最优运输问题。

快速求解最优输运问题：往返法

A fast approach to optimal transport: The back-and-forth method

We study the control of a linear dynamical system with adversarial
disturbances (as opposed to statistical noise). The objective we consider is
one of regret: we desire an online control procedure that can do nearly as well
as that of a procedure that has full knowledge of the disturbances in
hindsight. Our main result is an efficient algorithm that provides nearly tight
regret bounds for this problem. From a technical standpoint, this work
generalizes upon previous work in two main aspects: our model allows for
adversarial noise in the dynamics, and allows for general convex costs.

本文研究带有敌对干扰的线性动态系统的控制，在几乎不知道扰动信息的情况下，实现近乎最优的在线控制过程，主要贡献是提出一种算法来提供几乎紧密的遗憾界，这一研究在技术层面上对以前的工作进行了推广和扩展。