Transportation of probability measures underlies many core tasks in statistics and machine learning, from variational inference to generative modeling. A typical goal is to represent a target probability measure of interest as the push-forward of a tractable source measure through a learned map. We present a new construction of such a transport map, given the ability to evaluate the score of the target distribution. Specifically, we characterize the map as a zero of an infinite-dimensional score-residual operator and derive a Newton-type method for iteratively constructing such a zero. We prove convergence of these iterations by invoking classical elliptic regularity theory for partial differential equations (PDE) and show that this construction enjoys rapid convergence, under smoothness assumptions on the target score. A key element of our approach is a generalization of the elementary Newton method to infinite-dimensional operators, other forms of which have appeared in nonlinear PDE and in dynamical systems. Our Newton construction, while developed in a functional setting, also suggests new iterative algorithms for approximating transport maps.

翻译：概率测度的迁移是统计与机器学习中许多核心任务的基础，从变分推断到生成建模。典型目标是通过一个学习到的映射，将易处理的源测度前推至感兴趣的目标概率测度。本文提出一种新的传输映射构造方法，该方法基于对目标分布得分的评估能力。具体而言，我们将该映射刻画为无穷维得分残差算子的零点，并推导出一种牛顿型迭代方法用于构造该零点。通过引入偏微分方程的经典椭圆正则化理论，我们证明了这些迭代的收敛性，并表明：在目标得分满足光滑性假设的条件下，该构造方法具有快速收敛性。本方法的一个关键要素是将基本牛顿法推广至无穷维算子——此类推广的其他形式已出现在非线性偏微分方程与动力系统中。尽管我们的牛顿构造是在函数框架下发展的，但也为逼近传输映射提供了新的迭代算法。