Related papers: Kinetic Optimal Transport (OTIKIN) -- Part 1: Seco…
In this paper, we extend the scope of Caffarelli's contraction theorem, which provides a measure of the Lipschitz constant for optimal transport maps between log-concave probability densities in $\R^d$. Our focus is on a broader category of…
The assignment problem, a cornerstone of operations research, seeks an optimal one-to-one mapping between agents and tasks to minimize total cost. This work traces its evolution from classical formulations and algorithms to modern optimal…
Computing optimal transport (OT) distances such as the earth mover's distance is a fundamental problem in machine learning, statistics, and computer vision. In this paper, we study the problem of approximating the general OT distance…
Multi-marginal optimal transport enables one to compare multiple probability measures, which increasingly finds application in multi-task learning problems. One practical limitation of multi-marginal transport is computational scalability…
Optimal transport distances (OT) have been widely used in recent work in Machine Learning as ways to compare probability distributions. These are costly to compute when the data lives in high dimension. Recent work by Paty et al., 2019,…
We formulate and solve a class of finite-time transport and mixing problems in the set-oriented framework. The aim is to obtain optimal discrete-time perturbations in nonlinear dynamical systems to transport a specified initial measure on…
Kinetic theory of dissipative particle dynamics is developed in terms of a Boltzmann pair collision theory. The kinetic transport coefficients are computed from explicit collision integrals and compared favourably with detailed simulations.…
Optimal transport theory, originally developed in the 18th century for civil engineering, has since become a powerful optimization framework across disciplines, from generative AI to cell biology. In physics, it has recently been shown to…
We develop a discrete optimal transport framework for analyzing simulated annealing algorithms on finite state spaces. Building on the discrete Wasserstein metric introduced by Maas (J. Funct. Anal., 2011), we define a generalized discrete…
In this paper we revisit a class of optimal transport problems associated to non-autonomous linear control systems. Building on properties of the cost functions on $\mathbb{R}^{d}\times\mathbb{R}^{d}$ derived from suitable variational…
Particle gradient descent, which uses particles to represent a probability measure and performs gradient descent on particles in parallel, is widely used to optimize functions of probability measures. This paper considers particle gradient…
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…
We study optimal transport between two high-dimensional distributions $\mu,\nu$ in $R^n$ from an algorithmic perspective: given $x \sim \mu$, find a close $y \sim \nu$ in $poly(n)$ time, where $n$ is the dimension of $x,y$. Thus, running…
This paper presents a novel two-step approach for the fundamental problem of learning an optimal map from one distribution to another. First, we learn an optimal transport (OT) plan, which can be thought as a one-to-many map between the two…
We prove quantitative bounds on the stability of optimal transport maps and Kantorovich potentials from a fixed source measure $\rho$ under variations of the target measure $\mu$, when the cost function is the squared Riemannian distance on…
In machine learning and computer graphics, a fundamental task is the approximation of a probability density function through a well-dispersed collection of samples. Providing a formal metric for measuring the distance between probability…
We consider optimal transportation of measures on metric and topological spaces in the case where the cost function and marginal distributions depend on a parameter with values in a metric space. The Hausdorff distance between the sets of…
The theory of Monge-Kantorovich Optimal Mass Transport (OMT) has in recent years spurred a fast developing phase of research in stochastic control, control of ensemble systems, thermodynamics, data science, and several other fields in…
In machine learning, Optimal Transport (OT) theory is extensively utilized to compare probability distributions across various applications, such as graph data represented by node distributions and image data represented by pixel…
Optimal transport maps define a one-to-one correspondence between probability distributions, and as such have grown popular for machine learning applications. However, these maps are generally defined on empirical observations and cannot be…