Related papers: Efficient Gradient Flows in Sliced-Wasserstein Spa…
Algorithms based on discretizing Langevin diffusion are popular tools for sampling from high-dimensional distributions. We develop novel connections between such Monte Carlo algorithms, the theory of Wasserstein gradient flow, and the…
The Wasserstein gradient flow structure of the PDE system governing multiphase flows in porous media was recently highlighted in [C. Canc\`es, T. O. Gallou\"et, and L. Monsaingeon, {\it Anal. PDE} 10(8):1845--1876, 2017]. The model can thus…
Solving Fredholm equations of the first kind is crucial in many areas of the applied sciences. In this work we adopt a probabilistic and variational point of view by considering a minimization problem in the space of probability measures…
This is an expository paper on the theory of gradient flows, and in particular of those PDEs which can be interpreted as gradient flows for the Wasserstein metric on the space of probability measures (a distance induced by optimal…
We analyze some parabolic PDEs with different drift terms which are gradient flows in the Wasserstein space and consider the corresponding discrete-in-time JKO scheme. We prove with optimal transport techniques how to control the L p and L…
As a counterpoint to recent numerical methods for crystal surface evolution, which agree well with microscopic dynamics but suffer from significant stiffness that prevents simulation on fine spatial grids, we develop a new numerical method…
A nonlinear diffusion equation, interpreted as a Wasserstein gradient flow, is numerically solved in one space dimension using a higher-order minimizing movement scheme based on the BDF (backward differentiation formula) discretization. In…
In this paper we bring together some of the key ideas and methods of two disparate fields of mathematical research, frame theory and optimal transport, using the methods of the second to answer questions posed in the first. In particular,…
While behavior cloning with flow/diffusion policies excels at learning complex skills from demonstrations, it remains vulnerable to distributional shift, and standard RL methods struggle to fine-tune these models due to their iterative…
Sampling from nonsmooth target probability distributions is essential in various applications, including the Bayesian Lasso. We propose a splitting-based sampling algorithm for the time-implicit discretization of the probability flow for…
We study the JKO scheme for the total variation, characterize the optimizers, prove some of their qualitative properties (in particular a form of maximum principle and in some cases, a minimum principle as well). Finally, we establish a…
We propose a variational finite volume scheme to approximate the solutions to Wasserstein gradient flows. The time discretization is based on an implicit linearization of the Wasserstein distance expressed thanks to Benamou-Brenier formula,…
The Gromov-Wasserstein distance is a notable extension of optimal transport. In contrast to the classic Wasserstein distance, it solves a quadratic assignment problem that minimizes the pair-wise distance distortion under the transportation…
Variational inference (VI) can be cast as an optimization problem in which the variational parameters are tuned to closely align a variational distribution with the true posterior. The optimization task can be approached through vanilla…
The JKO scheme provides the discrete-in-time approximation for the solutions of evolutionary equations with Wasserstein gradient structure. We study a natural space-discretization of this scheme by restricting the minimization to the…
We propose a projected Wasserstein gradient descent method (pWGD) for high-dimensional Bayesian inference problems. The underlying density function of a particle system of WGD is approximated by kernel density estimation (KDE), which faces…
We consider the approximation of initial/boundary value problems involving, possibly high-dimensional, dissipative evolution partial differential equations (PDEs) using a deep neural network framework. More specifically, we first propose…
We show that the continuous-time gradient descent in Rn can be viewed as an optimal controlled evolution for a suitable action functional; a similar result holds for stochastic gradient descent. We then provide an analogous characterization…
The Wasserstein space of probability measures is known for its intricate Riemannian structure, which underpins the Wasserstein geometry and enables gradient flow algorithms. However, the Wasserstein geometry may not be suitable for certain…
Efficient gradient computation of the Jacobian determinant term is a core problem in many machine learning settings, and especially so in the normalizing flow framework. Most proposed flow models therefore either restrict to a function…