Integrable Nonparametric Flows

TL;DR

This paper presents a novel nonparametric method for reconstructing infinitesimal normalizing flows from distribution perturbations, leveraging integrable vector fields and Green's functions, with potential applications in quantum Monte Carlo and machine learning.

Contribution

It introduces a nonparametric approach to reconstruct flows from distribution changes using integrable vector fields and Green's functions, reversing traditional flow learning.

Findings

Validated on low-dimensional problems

Flow reconstruction closely related to electrostatics

Potential applications in quantum Monte Carlo and machine learning

Abstract

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.