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We study unsupervised generative modeling in terms of the optimal transport (OT) problem between true (but unknown) data distribution $P_X$ and the latent variable model distribution $P_G$. We show that the OT problem can be equivalently…
Computing the empirical Wasserstein distance in the Wasserstein-distance-based independence test is an optimal transport (OT) problem with a special structure. This observation inspires us to study a special type of OT problem and propose a…
We study the problem of quantifying how far an empirical distribution deviates from Gaussianity under the framework of optimal transport. By exploiting the cone geometry of the relative translation invariant quadratic Wasserstein space, we…
Robust optimization aims to find optimum points from the collection of points that are feasible for every possible scenario of a given uncertain set. An optimum solution to a robust optimization problem is commonly found by the min-max…
The optimal transport (OT) map is a geometry-driven transformation between high-dimensional probability distributions which underpins a wide range of tasks in statistics, applied probability, and machine learning. However, existing…
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…
We consider Monge-Kantorovich optimal transport problems on $\mathbb{R}^d$, $d\ge 1$, with a convex cost function given by the cumulant generating function of a probability measure. Examples include the Wasserstein-2 transport whose cost…
Optimal Transport (OT) problem aims to find a transport plan that bridges two distributions while minimizing a given cost function. OT theory has been widely utilized in generative modeling. In the beginning, OT distance has been used as a…
This paper proposes a data-driven distributionally robust shortest path (DRSP) model where the distribution of the travel time in the transportation network can only be partially observed through a finite number of samples. Specifically, we…
This paper considers the problem of estimating a matrix that encodes pairwise distances in a finite metric space (or, more generally, the edge weight matrix of a network) under the barycentric coding model (BCM) with respect to the…
Sampling from diffusion probabilistic models (DPMs) can be viewed as a piecewise distribution transformation, which generally requires hundreds or thousands of steps of the inverse diffusion trajectory to get a high-quality image. Recent…
The Wasserstein metric or earth mover's distance (EMD) is a useful tool in statistics, machine learning and computer science with many applications to biological or medical imaging, among others. Especially in the light of increasingly…
In recent years, two prominent paradigms have shaped distributionally robust optimization (DRO), modeling distributional ambiguity through $\phi$-divergences and Wasserstein distances, respectively. While the former focuses on ambiguity in…
Many problems in dynamic data driven modeling deals with distributed rather than lumped observations. In this paper, we show that the Monge-Kantorovich optimal transport theory provides a unifying framework to tackle such problems in the…
We investigate the semi-discrete Optimal Transport (OT) problem, where a continuous source measure $\mu$ is transported to a discrete target measure $\nu$, with particular attention to the OT map approximation. In this setting, Stochastic…
We present a framework for embedding graph structured data into a vector space, taking into account node features and topology of a graph into the optimal transport (OT) problem. Then we propose a novel distance between two graphs, named…
We give a new and constructive proof of the existence of global-in-time weak solutions of the 3-dimensional incompressible semi-geostrophic equations (SG) in geostrophic coordinates, for arbitrary initial measures with compact support. This…
We study the use of amortized optimization to predict optimal transport (OT) maps from the input measures, which we call Meta OT. This helps repeatedly solve similar OT problems between different measures by leveraging the knowledge and…
This note gives a self-contained overview of some important properties of the Gromov-Wasserstein (GW) distance, compared with the standard linear optimal transport (OT) framework. More specifically, I explore the following questions: are GW…
This paper addresses the practical challenge in Entropic Optimal Transport (EOT) where the underlying ground cost function is typically latent and unobserved. Rather than assuming a fixed geometric cost, we adopt a data-driven approach…