In the theory of optimal transport, the Knothe-Rosenblatt (KR) rearrangement provides an explicit construction to map between two probability measures by building one-dimensional transformations from the marginal conditionals of one measure to the other. The KR map has shown to be useful in different realms of mathematics and statistics, from proving functional inequalities to designing methodologies for sampling conditional distributions. It is known that the KR rearrangement can be obtained as the limit of a sequence of optimal transport maps with a weighted quadratic cost. We extend these results in this work by showing that one can obtain the KR map as a limit of maps that solve a relaxation of the weighted-cost optimal transport problem with a soft-constraint for the target distribution. In addition, we show that this procedure also applies to the construction of triangular velocity fields via dynamic optimal transport yielding optimal velocity fields. This justifies various variational methodologies for estimating KR maps in practice by minimizing a divergence between the target and pushforward measure through an approximate map. Moreover, it opens the possibilities for novel static and dynamic OT estimators for KR maps.

翻译：在最优化传输理论中，Knothe-Rosenblatt（KR）重排通过构建从一个概率分布的边缘条件分布到另一个分布的边缘条件分布的一维变换，提供了两个概率分布之间映射的显式构造方法。KR映射已被证明在数学与统计学的多个领域具有重要应用价值，从证明函数不等式到设计条件分布采样方法。已知KR重排可通过一系列具有加权二次成本的最优传输映射的极限获得。本研究扩展了这些结果，证明了KR映射可作为一类松弛加权成本最优传输问题解的极限获得，该问题对目标分布施加了软约束。此外，我们证明该方法同样适用于通过动态最优传输构建三角速度场，从而得到最优速度场。这为实践中通过最小化目标分布与近似映射推前分布之间的散度来估计KR映射的各种变分方法提供了理论依据，同时为KR映射的新型静态与动态最优传输估计器开辟了可能性。