Integrable Nonparametric Flows
This work provides a novel approach to reconstruct flows from distribution perturbations, which could be significant for researchers in quantum Monte Carlo and machine learning by offering a new tool for distribution manipulation.
This paper introduces a method to reconstruct an infinitesimal normalizing flow from a given infinitesimal change to a probability distribution. By choosing an integrable vector field, the problem becomes solvable via Green's functions, offering a nonparametric representation of the flow.
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.