Constructive conditional normalizing flows

• URL: http://arxiv.org/abs/2602.08606v1
• Date: Mon, 09 Feb 2026 12:52:47 GMT
• Title: Constructive conditional normalizing flows
• Authors: Borjan Geshkovski, Domènec Ruiz-Balet,
• Abstract summary: We consider the problem of simultaneously approximating $$ and the pushforward $_#$ by means of the flow of a continuity equation.<n>We provide an explicit construction based on a polar-like decomposition of the Lagrange interpolant of $$.
• Score: 0.8594140167290097
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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 Knöthe-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.

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