通过软约束最优传输构造 Knothe-Rosenblatt 映射
Knothe-Rosenblatt maps via soft-constrained optimal transport
AI总结：
本文提出通过带软约束的加权代价最优传输松弛问题的极限来构造 Knothe-Rosenblatt 映射，并将其推广到动态最优传输中的三角速度场构造，为 KR 映射的静态与动态变分估计方法提供了理论依据。
摘要：
在最优传输理论中，Knothe-Rosenblatt（KR）重排提供了一种显式构造，通过将一个测度的边际条件分布的一维变换映射到另一个测度，从而在两个概率测度之间建立映射。KR 映射在数学和统计学的不同领域中已被证明是有用的，从证明函数不等式到设计条件分布采样的方法。已知 KR 重排可以作为具有加权二次代价的最优传输映射序列的极限得到。我们在本工作中扩展了这些结果，证明可以通过求解带目标分布软约束的加权代价最优传输问题的松弛版本的映射序列的极限来得到 KR 映射。此外，我们证明该过程也适用于通过动态最优传输构造三角速度场，从而得到最优速度场。这为在实践中通过最小化目标测度与经近似映射推送得到的测度之间的散度来估计 KR 映射的各种变分方法提供了理论依据。此外，它为 KR 映射的新型静态和动态最优传输估计器开辟了可能性。
英文摘要：
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.
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