Related papers: Lipschitz Transport Maps via the Follmer Flow
We establish sufficient conditions for the existence of globally Lipschitz transport maps between probability measures and their log-Lipschitz perturbations, with dimension-free bounds. Our results include Gaussian measures on Euclidean…
We introduce a novel unit-time ordinary differential equation (ODE) flow called the preconditioned F\"{o}llmer flow, which efficiently transforms a Gaussian measure into a desired target measure at time 1. To discretize the flow, we apply…
This paper regroups some of the basic properties of Lipschitz maps and their flows. Many of the results presented here are classical in the case of smooth maps. We prove them here in the Lipschitz case for a better understanding of the…
Flow-based methods for sampling and generative modeling use continuous-time dynamical systems to represent a {transport map} that pushes forward a source measure to a target measure. The introduction of a time axis provides considerable…
Under general assumptions on the target distribution $p^\star$, we establish a sharp Lipschitz regularity theory for flow-matching vector fields and diffusion-model scores, with optimal dependence on time and dimension. As applications, we…
In this note, we propose a probabilistic approach to bound the (dimension-free) Lipschitz constant of the Langevin flow map on $\mathbb{R}^d$ introduced by Kim and Milman (2012). As example of application, we construct Lipschitz maps from a…
We prove new Lipschitz properties for transport maps along heat flows, constructed by Kim and Milman. For (semi)-log-concave measures and Gaussian mixtures, our bounds have several applications: eigenvalues comparisons, dimensional…
We consider the 1-harmonic flow of maps from a bounded domain into a submanifold of a Euclidean space, i.e. the gradient flow of the total variation functional restricted to maps taking values in the manifold. We restrict ourselves to…
The nonlinear wave equation $u_{tt}-c(u)(c(u)u_x)_x=0$ determines a flow of conservative solutions taking values in the space $H^1(\mathbb{R})$. However, this flow is not continuous w.r.t. the natural $H^1$ distance. Aim of this paper is to…
We show the existence of a global unique and analytic solution for the mean curvature flow, the surface diffusion flow and the Willmore flow of entire graphs for Lipschitz initial data with small Lipschitz norm. We also show the existence…
Consider in the phase space of classical mechanics a Radon measure that is a probability density carried by the graph of a Lipschitz continuous (or even less regular) vector field. We study the structure of the push-forward of such a…
A general theory is provided delivering convergence of maximal cyclically monotone mappings containing the supports of coupling measures of sequences of pairs of possibly random probability measures on Euclidean space. The theory is based…
Contraction properties of transport maps between probability measures play an important role in the theory of functional inequalities. The actual construction of such maps, however, is a non-trivial task and, so far, relies mostly on the…
Rademacher theorem states that every Lipschitz function on the Euclidean space is differentiable almost everywhere, where "almost everywhere" refers to the Lebesgue measure. In this paper we prove a differentiability result of similar type,…
We consider a one dimensional transport model with nonlocal velocity given by the Hilbert transform and develop a global well-posedness theory of probability measure solutions. Both the viscous and non-viscous cases are analyzed. Both in…
Normalizing flows have grown more popular over the last few years; however, they continue to be computationally expensive, making them difficult to be accepted into the broader machine learning community. In this paper, we introduce a…
According to a celebrated result of L. Caffarelli, every optimal mass transportation mapping pushing forward the standard Gaussian measure onto a log-concave measure $e^{-W} dx$ with $D^2 W \ge {Id}$ is 1-Lipschitz. We present a short…
The sliced-Wasserstein flow is an evolution equation where a probability density evolves in time, advected by a velocity field computed as the average among directions in the unit sphere of the optimal transport displacements from its 1D…
Many normalizing flow architectures impose regularity constraints, yet their distributional approximation properties are not fully characterized. We study the expressivity of bi-Lipschitz normalizing flows through the lens of score-based…
In Part I, we construct a class of examples of initial velocities for which the unique solution to the Euler equations in the plane has an associated flow map that lies in no Holder space of positive exponent for any positive time. In Part…