Monge-Ampère Flow for Generative Modeling
This work addresses generative modeling for unsupervised density estimation and variational calculations, introducing a novel integration of fluid dynamics and optimal transport into flow-based models, which is incremental in combining existing theories.
The paper tackles generative modeling by proposing Monge-Ampère flow, a deep generative model based on optimal transport theory, which achieves tractable likelihoods and efficient sampling, as demonstrated on MNIST density estimation and the 2D Ising model at the critical point.
We present a deep generative model, named Monge-Ampère flow, which builds on continuous-time gradient flow arising from the Monge-Ampère 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-Ampère 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-Ampère equation, optimal transport, and fluid dynamics into reversible flow-based generative models.