Related papers: A deep learning approach to multi-marginal optimal…
We propose deep learning methods for classical Monge's optimal mass transportation problems, where where the distribution constraint is treated as penalty terms defined by the maximum mean discrepancy in the theory of Hilbert space…
Monge map refers to the optimal transport map between two probability distributions and provides a principled approach to transform one distribution to another. Neural network based optimal transport map solver has gained great attention in…
We study the problem of estimating, in the sense of optimal transport metrics, a measure which is assumed supported on a manifold embedded in a Hilbert space. By establishing a precise connection between optimal transport metrics, optimal…
Optimal transport (OT) is a powerful geometric tool used to compare and align probability measures following the least effort principle. Despite its widespread use in machine learning (ML), OT problem still bears its computational burden,…
We establish numerical methods for solving the martingale optimal transport problem (MOT) - a version of the classical optimal transport with an additional martingale constraint on transport's dynamics. We prove that the MOT value can be…
In this paper, we consider the Monge optimal transport problem with distance cost. We prove that in some metric spaces, possibly with many branching geodesics, an optimal transport map exists if the first marginal is absolutely continuous.…
This paper presents a widely applicable approach to solving (multi-marginal, martingale) optimal transport and related problems via neural networks. The core idea is to penalize the optimization problem in its dual formulation and reduce it…
We propose two deep neural network-based methods for solving semi-martingale optimal transport problems. The first method is based on a relaxation/penalization of the terminal constraint, and is solved using deep neural networks. The second…
We consider the Monge problem of optimal transport between a compactly supported source measure and a target probability measure with unbounded support. We consider the convergence of optimal maps and potential functions when the target…
In the absence of prior knowledge, ordinal embedding methods obtain new representation for items in a low-dimensional Euclidean space via a set of quadruple-wise comparisons. These ordinal comparisons often come from human annotators, and…
We propose a volumetric formulation for computing the Optimal Transport problem defined on surfaces in $\mathbb{R}^3$, found in disciplines like optics, computer graphics, and computational methodologies. Instead of directly tackling the…
In this work we present deep learning implementations of two popular theoretical constrained optimization algorithms in infinite dimensional Hilbert spaces, namely, the penalty and the augmented Lagrangian methods. We test these algorithms…
The dynamic formulation of optimal transport has attracted growing interests in scientific computing and machine learning, and its computation requires to solve a PDE-constrained optimization problem. The classical Eulerian discretization…
In this work, we construct a novel numerical method for solving the multi-marginal optimal transport problems with Coulomb cost. This type of optimal transport problems arises in quantum physics and plays an important role in understanding…
The problem of optimal mass transport arises in numerous applications including image registration, mesh generation, reflector design, and astrophysics. One approach to solving this problem is via the Monge-Amp\`ere equation. While recent…
We study multi-marginal optimal transport problems from a probabilistic graphical model perspective. We point out an elegant connection between the two when the underlying cost for optimal transport allows a graph structure. In particular,…
We study Neural Optimal Transport in infinite-dimensional Hilbert spaces. In non-regular settings, Semi-dual Neural OT often generates spurious solutions that fail to accurately capture target distributions. We analytically characterize…
We present a numerical method to solve the optimal transport problem with a quadratic cost when the source and target measures are periodic probability densities. This method is based on a numerical resolution of the corresponding…
We propose a numerical algorithm for the computation of multi-marginal optimal transport (MMOT) problems involving general probability measures that are not necessarily discrete. By developing a relaxation scheme in which marginal…
We introduce a formulation of optimal transport problem for distributions on function spaces, where the stochastic map between functional domains can be partially represented in terms of an (infinite-dimensional) Hilbert-Schmidt operator…