Transport map methods offer a powerful statistical learning tool that can couple a target high-dimensional random variable with some reference random variable using invertible transformations. This paper presents new computational techniques for building the Knothe--Rosenblatt (KR) rearrangement based on general separable functions. We first introduce a new construction of the KR rearrangement -- with guaranteed invertibility in its numerical implementation -- based on approximating the density of the target random variable using tensor-product spectral polynomials and downward closed sparse index sets. Compared to other constructions of KR arrangements based on either multi-linear approximations or nonlinear optimizations, our new construction only relies on a weighted least square approximation procedure. Then, inspired by the recently developed deep tensor trains (Cui and Dolgov, Found. Comput. Math. 22:1863--1922, 2022), we enhance the approximation power of sparse polynomials by preconditioning the density approximation problem using compositions of maps. This is particularly suitable for high-dimensional and concentrated probability densities commonly seen in many applications. We approximate the complicated target density by a composition of self-reinforced KR rearrangements, in which previously constructed KR rearrangements -- based on the same approximation ansatz -- are used to precondition the density approximation problem for building each new KR rearrangement. We demonstrate the efficiency of our proposed methods and the importance of using the composite map on several inverse problems governed by ordinary differential equations (ODEs) and partial differential equations (PDEs).

翻译：传输映射方法提供了一种强大的统计学习工具，可通过可逆变换将高维目标随机变量与某个参考随机变量耦合。本文提出基于一般可分离函数构建 Knothe-Rosenblatt (KR) 重排的新型计算技术。我们首先引入了一种新的 KR 重排构造方法——在其数值实现中保证了可逆性——该方法基于使用张量积谱多项式和向下封闭稀疏指标集来逼近目标随机变量的密度。与基于多线性逼近或非线性优化的其他 KR 重排构造相比，我们的新构造仅依赖于加权最小二乘逼近过程。然后，受近期发展的深度张量列（Cui 和 Dolgov，《Found. Comput. Math.》22:1863–1922, 2022）启发，我们通过使用映射的复合对密度逼近问题进行预处理，增强了稀疏多项式的逼近能力。这尤其适用于许多应用中常见的高维和集中概率密度。我们通过自强化 KR 重排的复合来逼近复杂的目标密度，其中先前构建的 KR 重排（基于相同的逼近假设）被用于预处理密度逼近问题，以构建每个新的 KR 重排。我们通过多个由常微分方程（ODE）和偏微分方程（PDE）控制的反问题，展示了所提出方法的高效性以及使用复合映射的重要性。