Approximations of the cumulative distribution function using transport maps learning

This paper considers approximating the cumulative distribution function (CDF). For many important probability distributions, such as the normal distribution, their CDFs lack closed-form expressions representable by elementary functions. Although approximation methods exist, common techniques such as the empirical CDF typically rely on large amounts of sample data to construct sufficiently accurate approximations. The aim of this paper is to provide accurate and data-efficient closed-form approximations for CDFs. Our methodology is inspired by the theory of transport maps. We leverage the fundamental property that in the specific one-dimensional case, the transport map transforming a target random variable to the standard uniform distribution U(0,1) is identical to the target variable's CDF. Building upon this key insight, we propose Transport Map Learning (TML). We utilize TML to train a neural network whose output is subsequently processed by a sigmoid function. This composite architecture serves as our closed-form CDF approximation, inherently constraining the output to the [0,1] range appropriate for a CDF. The effectiveness of the proposed method is validated on three benchmark probability distributions: the standard normal distribution, the beta distribution, and the gamma distribution. The results demonstrate that, given the same amount of training data, the proposed TML method generates highly accurate closed-form approximations for the CDFs. These approximations achieve superior accuracy compared to established methods based on the empirical CDF combined with various interpolation strategies.