Integration against, and hence sampling from, high-dimensional probability distributions is of essential importance in many application areas and has been an active research area for decades. One approach that has drawn increasing attention in recent years has been the generation of samples from a target distribution $\mathbb{P}_{\mathrm{tar}}$ using transport maps: if $\mathbb{P}_{\mathrm{tar}} = T_\# \mathbb{P}_{\mathrm{ref}}$ is the pushforward of an easily-sampled probability distribution $\mathbb{P}_{\mathrm{ref}}$ under the transport map $T$, then the application of $T$ to $\mathbb{P}_{\mathrm{ref}}$-distributed samples yields $\mathbb{P}_{\mathrm{tar}}$-distributed samples. This paper proposes the application of transport maps not just to random samples, but also to quasi-Monte Carlo points, higher-order nets, and sparse grids in order for the transformed samples to inherit the original convergence rates that are often better than $N^{-1/2}$, $N$ being the number of samples/quadrature nodes. Our main result is the derivation of an explicit transport map for the case that $\mathbb{P}_{\mathrm{tar}}$ is a mixture of simple distributions, e.g.\ a Gaussian mixture, in which case application of the transport map $T$ requires the solution of an \emph{explicit} ODE with \emph{closed-form} right-hand side. Mixture distributions are of particular applicability and interest since many methods proceed by first approximating $\mathbb{P}_{\mathrm{tar}}$ by a mixture and then sampling from that mixture (often using importance reweighting). Hence, this paper allows for the sampling step to provide a better convergence rate than $N^{-1/2}$ for all such methods.

翻译：针对高维概率分布的积分及采样问题，在众多应用领域中具有根本重要性，且数十年来一直是活跃的研究领域。近年来，一种日益受到关注的方法是通过传输映射从目标分布$\mathbb{P}_{\mathrm{tar}}$中生成样本：若$\mathbb{P}_{\mathrm{tar}} = T_\# \mathbb{P}_{\mathrm{ref}}$是传输映射$T$下易采样分布$\mathbb{P}_{\mathrm{ref}}$的前推，则对$\mathbb{P}_{\mathrm{ref}}$分布样本施加$T$后即可获得$\mathbb{P}_{\mathrm{tar}}$分布样本。本文提出将传输映射不仅应用于随机样本，还适用于拟蒙哥马利点、高阶网及稀疏网格，使变换后的样本继承原始收敛速率（该速率通常优于$N^{-1/2}$，其中$N$为样本/求积节点数）。我们的主要成果是为$\mathbb{P}_{\mathrm{tar}}$为简单分布混合物（如高斯混合模型）的情况推导出显式传输映射，此时对传输映射$T$的求解需通过一个右端项具有闭式解形式的显式常微分方程。混合分布具有特殊适用价值，因为多数方法首先通过混合分布近似$\mathbb{P}_{\mathrm{tar}}$，再对该混合分布进行采样（常借助重要性重加权）。因此，本文使得所有此类方法的采样步骤均可获得优于$N^{-1/2}$的收敛速率。