Constructive conditional normalizing flows

arXiv:2602.08606v2 Announce Type: replace-cross Abstract: Motivated by applications in conditional sampling, given a probability measure mu and a diffeomorphism phi, we consider the problem of simultaneously approximating phi and the pushforward phi_{#}mu 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 phi. 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 phi -- such as the Knothe-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.

Source: arXiv cs.LG | 2026-05-12