相关论文: No-Regret Generative Modeling via Parabolic Monge-…

We present a deep generative model, named Monge-Amp\`ere flow, which builds on continuous-time gradient flow arising from the Monge-Amp\`ere equation in optimal transport theory. The generative map from the latent space to the data space…

Generative modelling is a key tool in unsupervised machine learning which has achieved stellar success in recent years. Despite this huge success, even the best generative models such as Generative Adversarial Networks (GANs) and…

This work builds the connection between the regularity theory of optimal transportation map, Monge-Amp\`{e}re equation and GANs, which gives a theoretic understanding of the major drawbacks of GANs: convergence difficulty and mode collapse.…

We show that the discrete Sinkhorn algorithm - as applied in the setting of Optimal Transport on a compact manifold - converges to the solution of a fully non-linear parabolic PDE of Monge-Ampere type, in a large-scale limit. The latter…

We consider a PDE approach to numerically solving the optimal transportation problem on the sphere. We focus on both the traditional squared geodesic cost and a logarithmic cost, which arises in the reflector antenna design problem. At each…

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…

We consider the numerical solution of the optimal transport problem between densities that are supported on sets of unequal dimension. Recent work by McCann and Pass reformulates this problem into a non-local Monge-Amp\`ere type equation.…

We propose a \textbf{uni}fied \textbf{f}ramework for \textbf{i}mplicit \textbf{ge}nerative \textbf{m}odeling (UnifiGem) with theoretical guarantees by integrating approaches from optimal transport, numerical ODE, density-ratio…

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…

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…

Diffusion models often yield highly curved trajectories and noisy score targets due to an uninformative, memoryless forward process that induces independent data-noise coupling. We propose Adjoint Schr\"odinger Bridge Matching (ASBM), a…

In many scientific fields imaging is used to relate a certain physical quantity to other dependent variables. Therefore, images can be considered as a map from a real-world coordinate system to the non-negative measurements being acquired.…

We address the numerical solution via Galerkin type methods of the Monge-Amp\`ere equation with transport boundary conditions arising in optimal mass transport, geometric optics and computational mesh or grid movement techniques. This fully…

We consider inference (filtering) problems over probabilistic graphical models with aggregate data generated by a large population of individuals. We propose a new efficient belief propagation type algorithm over tree-structured graphs with…

This paper presents a new framework for analyzing and designing no-regret algorithms for dynamic (possibly adversarial) systems. The proposed framework generalizes the popular online convex optimization framework and extends it to its…

We study the asymptotic behavior of solutions to the second boundary value problem for a parabolic PDE of Monge-Amp\`ere type arising from optimal mass transport. Our main result is an exponential rate of convergence for solutions of this…

In this paper, we explore the fundamental role of the Monge-Amp\`ere equation in deep learning, particularly in the context of Boltzmann machines and energy-based models. We first review the structure of Boltzmann learning and its relation…

Mixture-of-experts (MoE) models provide a flexible statistical framework for modeling heterogeneity and nonlinear relationships. In many modern applications, however, datasets are naturally distributed across multiple machines due to…

In moving mesh methods, the underlying mesh is dynamically adapted without changing the connectivity of the mesh. We specifically consider the generation of meshes which are adapted to a scalar monitor function through equidistribution.…

Score-based diffusion models demonstrate superior performance in generative tasks but encounter fundamental bottlenecks in inverse problems due to the analytical intractability of the time-dependent likelihood score. To bridge this gap, we…