Related papers: Entropic Wasserstein Gradient Flows
We prove the convergence of an implicit time discretization for the one-phase Mullins-Sekerka equation, possibly with additional non-local repulsion, proposed in [F. Otto, Arch. Rational Mech. Anal. 141 (1998) 63--103]. Our simple argument…
Recently, optimization on the Riemannian manifold have provided valuable insights to the optimization community. In this regard, extending these methods to to the Wasserstein space is of particular interest, since optimization on…
Gradient flows are a powerful tool for optimizing functionals in general metric spaces, including the space of probabilities endowed with the Wasserstein metric. A typical approach to solving this optimization problem relies on its…
The Poisson-Nernst-Planck system of equations used to model ionic transport is interpreted as a gradient flow for the Wasserstein distance and a free energy in the space of probability measures with finite second moment. A variational…
In this study, we investigate the performance of two novel first-order optimization algorithms, namely the rescaled-gradient flow (RGF) and the signed-gradient flow (SGF). These algorithms are derived from the forward Euler discretization…
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 paper investigates a time discrete variational model for splines in Wasserstein spaces to interpolate probability measures. Cubic splines in Euclidean space are known to minimize the integrated squared acceleration subject to a set of…
The possibility of using the Eulerian discretization for the problem of modelling high-dimensional distributions and sampling, is studied. The problem is posed as a minimization problem over the space of probability measures with respect to…
In this article, we formulate topology optimization problems concerning the mass distribution as minimization problems for functionals on the Wasserstein space. We relax optimization problems regarding non-convex objective functions on the…
Starting from Brenier's relaxed formulation of the incompressible Euler equation in terms of geodesics in the group of measure-preserving diffeomorphisms, we propose a numerical method based on Sinkhorn's algorithm for the entropic…
This work studies finite sample approximations of the exact and entropic regularized Wasserstein distances between centered Gaussian processes and, more generally, covariance operators of functional random processes. We first show that…
Distance measures between graphs are important primitives for a variety of learning tasks. In this work, we describe an unsupervised, optimal transport based approach to define a distance between graphs. Our idea is to derive…
On the space of probability densities, we extend the Wasserstein geodesics to the case of higher-order interpolation such as cubic spline interpolation. After presenting the natural extension of cubic splines to the Wasserstein space, we…
The Wasserstein distance received a lot of attention recently in the community of machine learning, especially for its principled way of comparing distributions. It has found numerous applications in several hard problems, such as domain…
Sampling a target probability distribution with an unknown normalization constant is a fundamental challenge in computational science and engineering. Recent work shows that algorithms derived by considering gradient flows in the space of…
We introduce Deep Set Linearized Optimal Transport, an algorithm designed for the efficient simultaneous embedding of point clouds into an $L^2-$space. This embedding preserves specific low-dimensional structures within the Wasserstein…
This paper studies iterative schemes for measure transfer and approximation problems, which are defined through a slicing-and-matching procedure. Similar to the sliced Wasserstein distance, these schemes benefit from the availability of…
We give a comprehensive description of Wasserstein gradient flows of maximum mean discrepancy (MMD) functionals $\mathcal F_\nu := \text{MMD}_K^2(\cdot, \nu)$ towards given target measures $\nu$ on the real line, where we focus on the…
This paper provides results on Wasserstein gradient flows between measures on the real line. Utilizing the isometric embedding of the Wasserstein space $\mathcal P_2(\mathbb R)$ into the Hilbert space $L_2((0,1))$, Wasserstein gradient…
The purpose of this paper is to answer a few open questions in the interface of kernel methods and PDE gradient flows. Motivated by recent advances in machine learning, particularly in generative modeling and sampling, we present a rigorous…