Related papers: A Computational Framework for Solving Wasserstein …
This paper reviews different numerical methods for specific examples of Wasserstein gradient flows: we focus on nonlinear Fokker-Planck equations,but also discuss discretizations of the parabolic-elliptic Keller-Segel model and of the…
We study a variant of the dynamical optimal transport problem in which the energy to be minimised is modulated by the covariance matrix of the distribution. Such transport metrics arise naturally in mean-field limits of certain ensemble…
Variational problems that involve Wasserstein distances and more generally optimal transport (OT) theory are playing an increasingly important role in data sciences. Such problems can be used to form an examplar measure out of various…
A new pairwise cost function is proposed for the optimal transport barycenter problem, adopting the form of the minimal action between two points, with a Lagrangian that takes into account an underlying probability distribution. Under this…
In this paper, we introduce a neural network-based method to address the high-dimensional dynamic unbalanced optimal transport (UOT) problem. Dynamic UOT focuses on the optimal transportation between two densities with unequal total mass,…
We propose a new formulation and learning strategy for computing the Wasserstein geodesic between two probability distributions in high dimensions. By applying the method of Lagrange multipliers to the dynamic formulation of the optimal…
We develop an efficient method for solving non-convex constrained optimization problems that are pervasive in economics. The optimal solution to these problems often involves randomization. We employ a Lagrangian framework and prove that…
This note outlines a mean-field approach to dynamic optimal transport problems based on the recently proposed McKean-Pontryagin maximum principle. Key aspects of the proposed methodology include i) avoidance of sampling over stochastic…
A material-based, i.e., Lagrangian, methodology for exact integration of flux by volume-preserving flows through a surface has been developed recently in [Karrasch, SIAM J. Appl. Math., 76 (2016), pp. 1178-1190]. In the present paper, we…
This article details a novel numerical scheme to approximate gradient flows for optimal transport (i.e. Wasserstein) metrics. These flows have proved useful to tackle theoretically and numerically non-linear diffusion equations that model…
We propose dynamical optimal transport (OT) problems constrained in a parameterized probability subset. In application problems such as deep learning, the probability distribution is often generated by a parameterized mapping function. In…
We study optimization problems whereby the optimization variable is a probability measure. Since the probability space is not a vector space, many classical and powerful methods for optimization (e.g., gradients) are of little help. Thus,…
The dynamical formulation of optimal transport, also known as Benamou-Brenier formulation or Computational Fluid Dynamics formulation, amounts to write the optimal transport problem as the optimization of a convex functional under a PDE…
Several problems in machine learning are naturally expressed as the design and analysis of time-evolving probability distributions. This includes sampling via diffusion methods, optimizing the weights of neural networks, and analyzing the…
This paper introduces a dynamic formulation of divergence-regularized optimal transport with weak targets on the path space. In our formulation, the classical relative entropy penalty is replaced by a general convex divergence, and terminal…
Trajectory optimization is an efficient approach for solving optimal control problems for complex robotic systems. It relies on two key components: first the transcription into a sparse nonlinear program, and second the corresponding solver…
Lagrangian relaxation is a versatile mathematical technique employed to relax constraints in an optimization problem, enabling the generation of dual bounds to prove the optimality of feasible solutions and the design of efficient…
Entropy-regularized optimal transport, which has strong links to the Schr\"odinger bridge problem in statistical mechanics, enjoys a variety of applications from trajectory inference to generative modeling. A major driver of renewed…
We propose a model of optimal parallel transport between vector fields on a connection graph, which consists of a weighted graph along with a map from its edges to an orthogonal group. Inspired by the well-known equivalence of 1-Wasserstein…
A data-driven formulation of the optimal transport problem is presented and solved using adaptively refined meshes to decompose the problem into a sequence of finite linear programming problems. Both the marginal distributions and their…