From Knothe's transport to Brenier's map and a continuation method for optimal transport

From Knothe's rearrangement to Brenier's optimal transport map

The Brenier optimal map and the Knothe-Rosenblatt rearrangement are two instances of a transport map, that is to say a map sending one measure onto another. The main interest of the former is that it solves the Monge-Kantorovich optimal transport problem, while the latter is very easy to compute, being given by an explicit formula. A few years ago, Carlier, Galichon, and Santambrogio showed that the Knothe rearrangement could be seen as the limit of the Brenier map when the quadratic cost degenerates. In this paper, we prove that on the torus (to avoid boundary issues), when all the data are s…

Causal transport in discrete time and applications

Loosely speaking, causal transport plans are a relaxation of adapted processes in the same sense as Kantorovich transport plans extend Monge-type transport maps. The corresponding causal version of the transport problem has recently been introduced by Lassalle. Working in a discrete time setup, we establish a dynamic programming principle that links the causal transport problem to the transport problem for general costs recently considered by Gozlan et al. Based on this recursive principle, we give conditions under which the celebrated Knothe-Rosenblatt rearrangement can be viewed as a causal …

适应性最优传输在过滤高斯过程之间

Adapted Optimal Transport between Filtered Gaussian Processes

本文研究了离散时间高斯设定下的适应性最优传输，引入了由高斯白噪声驱动的过滤高斯过程空间，并探讨了适应性2-Wasserstein距离的变分表示及高斯分布路径空间的完成。