# Integrable Nonparametric Flows

arXiv: https://arxiv.org/abs/2012.02035 · HTML: https://mlsanity.com/paper/2012.02035/

## Key facts

- Cited by 3 papers in the map (data through 13 Oct 2026).
- First posted on arXiv 3 Dec 2020.

## Built on by (most cited first, 3 of 3)

- [Learning Interpolations between Boltzmann Densities](https://mlsanity.com/paper/2301.07388/) (2023)
- [Numerical and geometrical aspects of flow-based variational quantum Monte Carlo](https://mlsanity.com/paper/2203.14824/) (2022)
- [Deformations of Boltzmann Distributions](https://mlsanity.com/paper/2210.13772/) (2022)

## Abstract (arXiv)

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.