Smooth transport map via diffusion process
Arthur Stéphanovitch
TL;DR
The paper develops a diffusion-based framework to construct smooth transport maps from the Gaussian measure to log Hölder perturbations, proving that the Langevin transport map achieves Hölder regularity $C^{\beta+1}$ up to a logarithmic factor when the perturbation $a$ satisfies Hölder-type regularity. It establishes the equivalence between the Föllmer and Langevin transports, provides forward-time formulations, and derives detailed derivative bounds via a Gronwall-type analysis and Faà di Bruno calculus. The results are extended to diffeomorphisms and ball-supported densities, with a Lusin-type relaxation that preserves high regularity on large mass subsets. Applications include transferring generalized log-Sobolev inequalities, achieving minimax-rate density estimation with GANs in non-compact domains, and informing score-based diffusion models for improved stability and accuracy.
Abstract
We extend the classical regularity theory of optimal transport to non-optimal transport maps generated by heat flow for perturbations of Gaussian measures. Considering probability measures of the form $dμ(x) = \exp\left(-\frac{|x|^2}{2} + a(x)\right)dx$ on $\mathbb{R}^d$ where $a$ has Hölder regularity $C^β$ with $β\geq 0$; we show that the Langevin map transporting the $d$-dimensional Gaussian distribution onto $μ$ achieves Hölder regularity $C^{β+ 1}$, up to a logarithmic factor. We additionally present applications of this result to functional inequalities and generative modelling.