🤖 AI Summary
This work proposes an explicit construction method based on continuity equation flows to address the challenge of simultaneously approximating the target diffeomorphic map and its pushforward distribution in conditional sampling. By leveraging Lagrangian interpolation and polar decomposition, the velocity field is decoupled into compressible and incompressible components: the former is realized via gradients of convex functions, while the latter is modeled by shear flows parameterized through perceptron networks with piecewise-constant weights. Combined with Maurey’s empirical method, this approach enables efficient joint approximation of both the mapping and the distribution in high dimensions, circumventing the exponential growth of weight discontinuities with dimensionality that plagues conventional methods. Consequently, it achieves significantly improved scalability and approximation accuracy.
📝 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 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.