We propose a general framework to robustly characterize joint and conditional probability distributions via transport maps. Transport maps or "flows" deterministically couple two distributions via an expressive monotone transformation. Yet, learning the parameters of such transformations in high dimensions is challenging given few samples from the unknown target distribution, and structural choices for these transformations can have a significant impact on performance. Here we formulate a systematic framework for representing and learning monotone maps, via invertible transformations of smooth functions, and demonstrate that the associated minimization problem has a unique global optimum. Given a hierarchical basis for the appropriate function space, we propose a sample-efficient adaptive algorithm that estimates a sparse approximation for the map. We demonstrate how this framework can learn densities with stable generalization performance across a wide range of sample sizes on real-world datasets.

中文翻译：

---

用于联合和条件密度估计的自适应传输框架

我们提出了一个通用框架，通过传输图稳健地表征联合和条件概率分布。传输图或“流”通过富有表现力的单调变换确定性地耦合两个分布。然而，鉴于来自未知目标分布的样本很少，在高维度上学习此类转换的参数具有挑战性，并且这些转换的结构选择可能会对性能产生重大影响。在这里，我们通过平滑函数的可逆变换制定了一个用于表示和学习单调映射的系统框架，并证明相关的最小化问题具有唯一的全局最优。给定适当功能空间的分层基础，我们提出了一种样本高效的自适应算法，可以估计地图的稀疏近似值。我们展示了该框架如何在真实世界数据集上的各种样本大小中以稳定的泛化性能学习密度。