Monge-Amp\`ere Flow for Generative Modeling

TL;DR

This paper introduces Monge-Ampère flow, a novel deep generative model based on optimal transport and fluid dynamics, enabling efficient sampling, likelihood computation, and symmetry constraints, demonstrated on image and physics datasets.

Contribution

It presents a new generative modeling approach leveraging Monge-Ampère equations and optimal control, integrating fluid dynamics for improved density estimation and inference.

Findings

Effective density estimation on MNIST.

Accurate variational calculations for the 2D Ising model.

Supports symmetry constraints in generative modeling.

Abstract

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 follows a dynamical system, where a learnable potential function guides a compressible fluid to flow towards the target density distribution. Training of the model amounts to solving an optimal control problem. The Monge-Amp\`ere flow has tractable likelihoods and supports efficient sampling and inference. One can easily impose symmetry constraints in the generative model by designing suitable scalar potential functions. We apply the approach to unsupervised density estimation of the MNIST dataset and variational calculation of the two-dimensional Ising model at the critical point. This approach brings insights and techniques from Monge-Amp\`ere equation, optimal transport, and fluid dynamics into reversible flow-based generative models.