# Constructive conditional normalizing flows

arXiv: https://arxiv.org/abs/2602.08606 · HTML: https://mlsanity.com/paper/2602.08606/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- First posted on arXiv 9 Feb 2026.

## Abstract (arXiv)

Motivated by applications in conditional sampling, given a probability measure μ and a diffeomorphism φ, we consider the problem of simultaneously approximating φ and the pushforward φ#μ by means of the flow of a continuity equation whose velocity field is a perceptron neural network with piecewise constant weights. We provide an explicit construction based on a polar-like decomposition of the Lagrange interpolant of φ. The latter involves a compressible component, given by the gradient of a particular convex function, which can be realized exactly, and an incompressible component, which -- after approximating via permutations -- can be implemented through shear flows intrinsic to the continuity equation. For more regular maps φ -- such as the Kn"othe-Rosenblatt rearrangement -- we provide an alternative, probabilistic construction inspired by the Maurey empirical method, in which the number of discontinuities in the weights doesn't scale inversely with the ambient dimension.