相关论文: Monge-Amp\`ere Flow for Generative Modeling

By building upon the recent theory that established the connection between implicit generative modeling (IGM) and optimal transport, in this study, we propose a novel parameter-free algorithm for learning the underlying distributions of…

We propose an Euler particle transport (EPT) approach for generative learning. The proposed approach is motivated by the problem of finding an optimal transport map from a reference distribution to a target distribution characterized by the…

Sampling from high-dimensional distributions is a fundamental problem in statistical research and practice. However, great challenges emerge when the target density function is unnormalized and contains isolated modes. We tackle this…

The inverse reflector problem arises in geometrical nonimaging optics: Given a light source and a target, the question is how to design a reflecting free-form surface such that a desired light density distribution is generated on the…

This paper introduces a fast and robust iterative scheme for the elliptic Monge-Amp\`ere equation with Dirichlet boundary conditions. The Monge-Amp\`ere equation is a nonlinear and degenerate equation, with applications in optimal…

This paper introduces a novel neural network-based approach to solving the Monge-Amp\`ere equation with the transport boundary condition, specifically targeted towards optical design applications. We leverage multilayer perceptron networks…

A fast method is presented for adaptive moving mesh generation in multi-dimensions using a domain decomposition parabolic Monge-Amp\`ere approach. The domain decomposition procedure employed here is non-iterative and involves splitting the…

A Monge-Amp\`ere (MA) equation arises when seeking an optimally transported mesh that equidistributes a given monitor function in Cartesian space. This MA equation is a fully nonlinear PDE, with a source term that is a function of the…

We study the estimation of optimal transport (OT) maps between an arbitrary source probability measure and a log-concave target probability measure. Our contributions are twofold. First, we propose a new evolution equation in the set of…

Generative AI (GenAI) has revolutionized data-driven modeling by enabling the synthesis of high-dimensional data across various applications, including image generation, language modeling, biomedical signal processing, and anomaly…

We propose a generative multivariate posterior sampler via flow matching. It offers a simple training objective, and does not require access to likelihood evaluation. The method learns a dynamic, block-triangular velocity field in the joint…

Generative models have achieved remarkable progress with the emergence of flow matching (FM). It has demonstrated strong generative capabilities and attracted significant attention as a simulation-free flow-based framework capable of…

We investigate a matrix dynamical system related to optimal mass transport in the linear category, namely, the problem of finding an optimal invertible matrix by which two covariance matrices are congruent. We first review the differential…

We propose a potential flow generator with $L_2$ optimal transport regularity, which can be easily integrated into a wide range of generative models including different versions of GANs and flow-based models. We show the correctness and…

Normalizing Flows (NF) are powerful likelihood-based generative models that are able to trade off between expressivity and tractability to model complex densities. A now well established research avenue leverages optimal transport (OT) and…

We identify a novel connection between a recently introduced pseudo-Riemannian framework for optimal mass transport and the geometry of Monge-Amp\`ere equations. We show this correspondence by application to an example from geophysical…

In recent works - both experimental and theoretical - it has been shown how to use computational geometry to efficently construct approximations to the optimal transport map between two given probability measures on Euclidean space, by…

Monge map refers to the optimal transport map between two probability distributions and provides a principled approach to transform one distribution to another. Neural network based optimal transport map solver has gained great attention in…

Autoregressive models are among the best performing neural density estimators. We describe an approach for increasing the flexibility of an autoregressive model, based on modelling the random numbers that the model uses internally when…

We study the most common image and informal description of the optimal transport problem for quadratic cost, also known as the second boundary value problem for the Monge--Amp\`{e}re equation -- What is the most efficient way to fill a hole…