Score-based Generative Neural Networks for Large-Scale Optimal Transport

TL;DR

This paper introduces a score-based generative model framework for learning regularized optimal transport couplings, enabling efficient sampling between high-dimensional distributions with theoretical convergence guarantees.

Contribution

It proposes a novel neural network approach to solve the Sinkhorn problem, with proven convergence and empirical success on large-scale tasks.

Findings

Effective sampling of optimal transport couplings in high dimensions

Convergence of neural network training for the Sinkhorn problem

Successful application to large-scale optimal transport tasks

Abstract

We consider the fundamental problem of sampling the optimal transport coupling between given source and target distributions. In certain cases, the optimal transport plan takes the form of a one-to-one mapping from the source support to the target support, but learning or even approximating such a map is computationally challenging for large and high-dimensional datasets due to the high cost of linear programming routines and an intrinsic curse of dimensionality. We study instead the Sinkhorn problem, a regularized form of optimal transport whose solutions are couplings between the source and the target distribution. We introduce a novel framework for learning the Sinkhorn coupling between two distributions in the form of a score-based generative model. Conditioned on source data, our procedure iterates Langevin Dynamics to sample target data according to the regularized optimal coupling. Key to this approach is a neural network parametrization of the Sinkhorn problem, and we prove convergence of gradient descent with respect to network parameters in this formulation. We demonstrate its empirical success on a variety of large scale optimal transport tasks.