Related papers: Gradient Flows for Frame Potentials on the Wassers…
We propose a mesoscopic modeling framework for optimal transportation networks with biological applications. The network is described in terms of a joint probability measure on the phase space of tensor-valued conductivity and position in…
We provide a framework to approximate the 2-Wasserstein distance and the optimal transport map, amenable to efficient training as well as statistical and geometric analysis. With the quadratic cost and considering the Kantorovich dual form…
Diffusion and flow-based generative models have achieved remarkable success in domains such as image synthesis, video generation, and natural language modeling. In this work, we extend these advances to weight space learning by leveraging…
This paper concerns the geometric structure of optimizers for frame potentials. We consider finite, real or complex frames and rotation or unitarily invariant potentials, and mostly specialize to Parseval frames, meaning the frame potential…
We prove a Pontryagin Maximum Principle for optimal control problems in the space of probability measures, where the dynamics is given by a transport equation with non-local velocity. We formulate this first-order optimality condition using…
Wasserstein Gradient Flow (WGF) describes the gradient dynamics of probability density within the Wasserstein space. WGF provides a promising approach for conducting optimization over the probability distributions. Numerically approximating…
Recent results have shown that for two-layer fully connected neural networks, gradient flow converges to a global optimum in the infinite width limit, by making a connection between the mean field dynamics and the Wasserstein gradient flow.…
Gaussian mixture models form a flexible and expressive parametric family of distributions that has found applications in a wide variety of applications. Unfortunately, fitting these models to data is a notoriously hard problem from a…
Optimal transport is widely used in pure and applied mathematics to find probabilistic solutions to hard combinatorial matching problems. We extend the Wasserstein metric and other elements of optimal transport from the matching of sets to…
Both the porous medium equation and the system of isentropic Euler equations can be considered as steepest descents on suitable manifolds of probability measures in the framework of optimal transport theory. By discretizing these…
We propose to study the Hessian metric of a functional on the space of probability measures endowed with the Wasserstein $2$-metric. We name it transport Hessian metric, which contains and extends the classical Wasserstein-$2$ metric. We…
We develop novel neural network-based implicit particle methods to compute high-dimensional Wasserstein-type gradient flows with linear and nonlinear mobility functions. The main idea is to use the Lagrangian formulation in the…
Applications of optimal transport have recently gained remarkable attention thanks to the computational advantages of entropic regularization. However, in most situations the Sinkhorn approximation of the Wasserstein distance is replaced by…
Many problems in machine learning can be formulated as solving entropy-regularized optimal transport on the space of probability measures. The canonical approach involves the Sinkhorn iterates, renowned for their rich mathematical…
We study policy gradient methods for continuous-action, entropy-regularized reinforcement learning through the lens of Wasserstein geometry. Starting from a Wasserstein proximal update, we derive Wasserstein Proximal Policy Gradient (WPPG)…
We develop a gradient-flow theory for time-dependent functionals defined in abstract metric spaces. Global well-posedness and asymptotic behavior of solutions are provided. Conditions on functionals and metric spaces allow to consider the…
Optimal transport has recently been brought forward as a tool for modeling and efficiently solving a variety of flow problems, such as origin-destination problems and multi-commodity flow problems. Although the framework has shown to be…
A comprehensive methodology for establishing the existence of gradient flows for cross-diffusion systems with respect to suitable energies is proposed. The approach is based on the construction of piecewise-in-time constant approximations…
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…
As a natural approach to modeling system safety conditions, chance constraint (CC) seeks to satisfy a set of uncertain inequalities individually or jointly with high probability. Although a joint CC offers stronger reliability certificate,…