Transportation of measure provides a versatile approach for modeling complex probability distributions, with applications in density estimation, Bayesian inference, generative modeling, and beyond. Monotone triangular transport maps—approximations of the Knothe–Rosenblatt (KR) rearrangement—are a canonical choice for these tasks. Yet the representation and parameterization of such maps have a significant impact on their generality and expressiveness, and on properties of the optimization problem that arises in learning a map from data (e.g., via maximum likelihood estimation). We present a general framework for representing monotone triangular maps via invertible transformations of smooth functions. We establish conditions on the transformation such that the associated infinite-dimensional minimization problem has no spurious local minima, i.e., all local minima are global minima; and we show for target distributions satisfying certain tail conditions that the unique global minimizer corresponds to the KR map. Given a sample from the target, we then propose an adaptive algorithm that estimates a sparse semi-parametric approximation of the underlying KR map. We demonstrate how this framework can be applied to joint and conditional density estimation, likelihood-free inference, and structure learning of directed graphical models, with stable generalization performance across a range of sample sizes.

测量传输提供了一种对复杂概率分布进行建模的通用方法，可应用于密度估计、贝叶斯推理、生成建模等领域。单调三角形传输图（Knothe-Rosenblatt (KR) 重排的近似值）是这些任务的典型选择。然而，此类映射的表示和参数化对其通用性和表达性以及从数据学习映射（例如，通过最大似然估计）时出现的优化问题的属性具有显着影响。我们提出了一个通过平滑函数的可逆变换来表示单调三角映射的通用框架。我们建立变换的条件，使得相关的无限维最小化问题没有虚假的局部最小值，即所有局部最小值都是全局最小值；我们证明，对于满足某些尾部条件的目标分布，唯一的全局最小化器对应于 KR 图。给定目标样本，我们提出一种自适应算法来估计底层 KR 图的稀疏半参数近似。我们演示了如何将该框架应用于联合和条件密度估计、无似然推理和有向图模型的结构学习，并在一系列样本大小上具有稳定的泛化性能。