Related papers: Monge-Amp\`ere Flow for Generative Modeling

This paper develops a fully discrete modified characteristic finite element method for a coupled system consisting of the fully nonlinear Monge-Amp\'ere equation and a transport equation. The system is the Eulerian formulation in the dual…

We formulate the inverse problem in a Bayesian framework and aim to train a generative model that allows us to simulate (i.e., sample from the likelihood) and do inference (i.e., sample from the posterior). We review the use of triangular…

We introduce and analyze a statistical estimator for Monge transport maps: solutions to the quadratic optimal transport problem in Euclidean space. For absolutely continuous source measures, this map is uniquely defined as the gradient of a…

This paper is about quantitative linearization results for the Monge-Amp\`ere equation with rough data. We develop a large-scale regularity theory and prove that if a measure $\mu$ is close to the Lebesgue measure in Wasserstein distance at…

We investigate the problem of sampling from posterior distributions with intractable normalizing constants in Bayesian inference. Our solution is a new generative modeling approach based on optimal transport (OT) that learns a deterministic…

Flow matching has recently emerged as a promising alternative to diffusion-based generative models, offering faster sampling and simpler training by learning continuous flows governed by ordinary differential equations. Despite growing…

We study the problem of training a flow-based generative model, parametrized by a two-layer autoencoder, to sample from a high-dimensional Gaussian mixture. We provide a sharp end-to-end analysis of the problem. First, we provide a tight…

The principles of mesh equidistribution and alignment play a fundamental role in the design of adaptive methods, and a metric tensor M and mesh metric are useful theoretical tools for understanding a methods level of mesh alignment, or…

We show how a symmetry reduction of the equations for incompressible hydrodynamics in three dimensions leads naturally to a Monge-Amp\`ere structure, and Burgers'-type vortices are a canonical class of solutions associated with this…

Maximum entropy modeling is a flexible and popular framework for formulating statistical models given partial knowledge. In this paper, rather than the traditional method of optimizing over the continuous density directly, we learn a smooth…

Normalizing flows are a powerful class of generative models for continuous random variables, showing both strong model flexibility and the potential for non-autoregressive generation. These benefits are also desired when modeling discrete…

We propose a unified, few-step generative modeling framework based on \emph{cumulative flow maps} for long-range transport in probability space, inspired by flow-map techniques for physical transport and dynamics. At its core is a…

Many data-driven decision problems are formulated using a nominal distribution estimated from historical data, while performance is ultimately determined by a deployment distribution that may be shifted, context-dependent, partially…

Sampling from high-dimensional, multi-modal distributions remains a fundamental challenge across domains such as statistical Bayesian inference and physics-based machine learning. In this paper, we propose Annealing Flow (AF), a method…

Finding a transformation between two unknown probability distributions from finite samples is crucial for modeling complex data distributions and performing tasks such as sample generation, domain adaptation and statistical inference. One…

We study the geometry of conditional optimal transport (COT) and prove a dynamical formulation which generalizes the Benamou-Brenier Theorem. Equipped with these tools, we propose a simulation-free flow-based method for conditional…

We propose a new framework for generative modeling based on a discrete-time stochastic control formulation of measure transport. Adapting classic results from control theory, we formulate our problem as a linear program whose dual variables…

Generative AI has achieved remarkable empirical success, but from the perspective of statistics it often remains opaque: its predictions may be accurate, yet the underlying mechanism is difficult to interpret, analyze, and trust. This book…

Repeatedly solving the parameterized optimal mass transport (pOMT) problem is a frequent task in applications such as image registration and adaptive grid generation. It is thus critical to develop a highly efficient reduced solver that is…

Flow-based generative models provide strong unconditional priors for inverse problems, but guiding their dynamics for conditional generation remains challenging. Recent work casts training-free conditional generation in flow models as an…