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$\unicode{x2014}$approximations of the Knothe$\unicode{x2013}$Rosenblatt (KR) rearrangement$\unicode{x2014}$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.

翻译：测度传输为复杂概率分布建模提供了一种通用方法，广泛应用于密度估计、贝叶斯推理、生成式建模等领域。单调三角传输映射——对Knothe–Rosenblatt (KR)重排的近似——是这些任务的经典选择。然而，这类映射的表示与参数化方式对其通用性、表达能力以及基于数据学习映射时（例如通过最大似然估计）所产生的优化问题性质具有重要影响。我们提出了一个通用框架，通过光滑函数的可逆变换来表示单调三角映射。我们建立了变换需满足的条件，以确保相关无限维最小化问题不存在虚假局部极小值，即所有局部极小值均为全局极小值；并证明对于满足特定尾部条件的目标分布，唯一全局极小值对应KR映射。基于目标分布的样本，我们进一步提出一种自适应算法，可估计底层KR映射的稀疏半参数近似。我们展示了该框架在联合密度与条件密度估计、无似然推理以及有向图模型结构学习中的应用，在不同样本量下均具有稳定的泛化性能。