Related papers: Long-Time Asymptotics of the Sliced-Wasserstein Fl…
Optimal transport distances, otherwise known as Wasserstein distances, have recently drawn ample attention in computer vision and machine learning as a powerful discrepancy measure for probability distributions. The recent developments on…
The sliced Wasserstein (SW) distances between two probability measures are defined as the expectation of the Wasserstein distance between two one-dimensional projections of the two measures. The randomness comes from a projecting direction…
This paper considers the problem of regression over distributions, which is becoming increasingly important in machine learning. Existing approaches often ignore the geometry of the probability space or are computationally expensive. To…
Tree-Sliced methods have recently emerged as an alternative to the traditional Sliced Wasserstein (SW) distance, replacing one-dimensional lines with tree-based metric spaces and incorporating a splitting mechanism for projecting measures.…
Many applications in machine learning involve data represented as probability distributions. The emergence of such data requires radically novel techniques to design tractable gradient flows on probability distributions over this type of…
The conventional sliced Wasserstein is defined between two probability measures that have realizations as vectors. When comparing two probability measures over images, practitioners first need to vectorize images and then project them to…
Optimal transport (\OT) theory defines a powerful set of tools to compare probability distributions. \OT~suffers however from a few drawbacks, computational and statistical, which have encouraged the proposal of several regularized variants…
Optimal Transport (OT) metrics allow for defining discrepancies between two probability measures. Wasserstein distance is for longer the celebrated OT-distance frequently-used in the literature, which seeks probability distributions to be…
We analyze the gradient flow of a potential energy in the space of probability measures when we substitute the optimal transport geometry with a geometry based on Sinkhorn divergences, a debiased version of entropic optimal transport. This…
In this paper we bring together some of the key ideas and methods of two disparate fields of mathematical research, frame theory and optimal transport, using the methods of the second to answer questions posed in the first. In particular,…
Many variants of the Wasserstein distance have been introduced to reduce its original computational burden. In particular the Sliced-Wasserstein distance (SW), which leverages one-dimensional projections for which a closed-form solution of…
In this paper, we study the stochastic Hamiltonian flow in Wasserstein manifold, the probability density space equipped with $L^2$-Wasserstein metric tensor, via the Wong--Zakai approximation. We begin our investigation by showing that the…
Optimal transport (OT) provides powerful tools for comparing probability measures in various types. The Wasserstein distance which arises naturally from the idea of OT is widely used in many machine learning applications. Unfortunately,…
The formation of cosmological large-scale structure is usually described in terms of the evolution of density fluctuations and their statistical measures, such as the power spectrum and correlation function. However, these statistics…
Rectified flow (Liu et al., 2022; Liu, 2022; Wu et al., 2023) is a method for defining a transport map between two distributions, and enjoys popularity in machine learning, although theoretical results supporting the validity of these…
Gromov--Wasserstein (GW) distances compare graphs, shapes, and point clouds through internal distances, without requiring a common coordinate system. This invariance is powerful, but discrete GW is a nonconvex quadratic optimal transport…
We study a system of drift-diffusion PDEs for a potentially infinite number of incompressible phases, subject to a joint pointwise volume constraint. Our analysis is based on the interpretation as a collection of coupled Wasserstein…
Wasserstein gradient flows are continuous time dynamics that define curves of steepest descent to minimize an objective function over the space of probability measures (i.e., the Wasserstein space). This objective is typically a divergence…
The Sliced Wasserstein (SW) distance has become a popular alternative to the Wasserstein distance for comparing probability measures. Widespread applications include image processing, domain adaptation and generative modelling, where it is…
Sliced Wasserstein distances are widely used in practice as a computationally efficient alternative to Wasserstein distances in high dimensions. In this paper, motivated by theoretical foundations of this alternative, we prove quantitative…