Generative adversarial networks (GANs) are an expressive class of neural generative models with tremendous success in modeling high-dimensional continuous measures. In this paper, we present a scalable method for unbalanced optimal transport (OT) based on the generative-adversarial framework. We formulate unbalanced OT as a problem of simultaneously learning a transport map and a scaling factor that push a source measure to a target measure in a cost-optimal manner. We provide theoretical justification for this formulation, showing that it is closely related to an existing static formulation by Liero et al. (2018). We then propose an algorithm for solving this problem based on stochastic alternating gradient updates, similar in practice to GANs, and perform numerical experiments demonstrating how this methodology can be applied to population modeling.
We propose new methodology for unbalanced optimal transport using generative adversarial networks. The authors consider the unbalanced optimal transport problem between two measures with different total mass using a stochastic min-max algorithm and local scaling The authors propose an approach to estimate unbalanced optimal transport between sampled measures that scales well in the dimension and in the number of samples. The paper introduces a static formulation for unbalanced optimal transport by learning simultaneously a transport map T and scaling factor xi.