相关论文: On the Regularity and Generalization of One-Step W…
Optimal transport provides an inherently geometric and highly structured framework for studying spaces of probability measures, supplying a rich theoretical toolkit for contemporary statistics, machine learning, and generative modelling. In…
This article presents a general approximation-theoretic framework to analyze measure transport algorithms for probabilistic modeling. A primary motivating application for such algorithms is sampling -- a central task in statistical…
We develop the theory of a metric, which we call the $\nu$-based Wasserstein metric and denote by $W_\nu$, on the set of probability measures $\mathcal P(X)$ on a domain $X \subseteq \mathbb{R}^m$. This metric is based on a slight…
Flow matching has recently emerged as a flexible and efficient framework for generative modelling by learning deterministic transport dynamics between probability measures. In this work, we extend flow matching to the space of probability…
Estimating Wasserstein distances between two high-dimensional densities suffers from the curse of dimensionality: one needs an exponential (wrt dimension) number of samples to ensure that the distance between two empirical measures is…
Given a family of probability measures in P(X), the space of probability measures on a Hilbert space X, our goal in this paper is to highlight one ore more curves in P(X) that summarize efficiently that family. We propose to study this…
This paper explores the problem of generative modeling, aiming to simulate diverse examples from an unknown distribution based on observed examples. While recent studies have focused on quantifying the statistical precision of popular…
Distributionally robust optimization has emerged as an attractive way to train robust machine learning models, capturing data uncertainty and distribution shifts. Recent statistical analyses have proved that generalization guarantees of…
Under general assumptions on the target distribution $p^\star$, we establish a sharp Lipschitz regularity theory for flow-matching vector fields and diffusion-model scores, with optimal dependence on time and dimension. As applications, we…
Flow-based methods for sampling and generative modeling use continuous-time dynamical systems to represent a {transport map} that pushes forward a source measure to a target measure. The introduction of a time axis provides considerable…
We propose the characteristic generator, a novel one-step generative model that combines the efficiency of sampling in Generative Adversarial Networks (GANs) with the stable performance of flow-based models. Our model is driven by…
We propose algorithms for sampling from posterior path measures $P(C([0, T], \mathbb{R}^d))$ under a general prior process. This leverages ideas from (1) controlled equilibrium dynamics, which gradually transport between two path measures,…
We study the Wasserstein natural gradient in parametric statistical models with continuous sample spaces. Our approach is to pull back the $L^2$-Wasserstein metric tensor in the probability density space to a parameter space, equipping the…
We develop a unified theoretical framework for data-free one-step sampling from unnormalized target distributions based on Wasserstein gradient flows. For a broad class of standard f-divergence objectives, we show that the induced velocity…
This work studies an explicit embedding of the set of probability measures into a Hilbert space, defined using optimal transport maps from a reference probability density. This embedding linearizes to some extent the 2-Wasserstein space,…
The notion of entropy-regularized optimal transport, also known as Sinkhorn divergence, has recently gained popularity in machine learning and statistics, as it makes feasible the use of smoothed optimal transportation distances for data…
This brief note aims to introduce the recent paradigm of distributional robustness in the field of shape and topology optimization. Acknowledging that the probability law of uncertain physical data is rarely known beyond a rough…
We introduce principal curves in Wasserstein space, and in general compact metric spaces. Our motivation for the Wasserstein case comes from optimal-transport-based trajectory inference, where a developing population of cells traces out a…
We propose a family of relaxations of the optimal transport problem which regularize the problem by introducing an additional minimization step over a small region around one of the underlying transporting measures. The type of…
Statistical solutions of incompressible Euler describe turbulent dynamics as time-parameterized laws on $L^2$ whose multi-point correlations satisfy an infinite hierarchy of weak identities. Modern generative samplers for PDE forecasting…