We introduce a method for reconstructing an infinitesimal normalizing flow given only an infinitesimal change to a (possibly unnormalized) probability distribution. This reverses the conventional task of normalizing flows -- rather than being given samples from a unknown target distribution and learning a flow that approximates the distribution, we are given a perturbation to an initial distribution and aim to reconstruct a flow that would generate samples from the known perturbed distribution. While this is an underdetermined problem, we find that choosing the flow to be an integrable vector field yields a solution closely related to electrostatics, and a solution can be computed by the method of Green's functions. Unlike conventional normalizing flows, this flow can be represented in an entirely nonparametric manner. We validate this derivation on low-dimensional problems, and discuss potential applications to problems in quantum Monte Carlo and machine learning.

翻译：我们引入了一种方法来重建一个极小的正常流, 仅仅给一个( 可能无法正常化的) 概率分布带来极小的微小变化。 这逆转了正常流的常规任务, 而不是从未知的目标分布中获得样本, 并学习出一个接近分布的流, 我们被给最初的分布带来扰动, 目的是重建一个从已知的扰动分布中产生样本的流。 虽然这是一个未下定义的问题, 我们发现选择流为一个可耐受控的矢量场, 会产生一种与电静剂密切相关的解决方案, 一种解决方案可以通过格林的功能方法来计算。 与常规的正常流不同, 这种流可以完全无分辨性的方式代表这种流。 我们验证这一在低维问题上的衍生, 并讨论对量的蒙特卡洛 和 机器学习中的问题的潜在应用 。