Constructive conditional normalizing flows
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
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2026
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| _version_ | 1866913115077083136 |
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| author | Geshkovski, Borjan Ruiz-Balet, Domènec |
| author_facet | Geshkovski, Borjan Ruiz-Balet, Domènec |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_08606 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Constructive conditional normalizing flows Geshkovski, Borjan Ruiz-Balet, Domènec Optimization and Control Machine Learning Analysis of PDEs Probability 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. |
| title | Constructive conditional normalizing flows |
| topic | Optimization and Control Machine Learning Analysis of PDEs Probability |
| url | https://arxiv.org/abs/2602.08606 |