  Optimal Transport (OT) naturally arises in many machine learning
applications, yet the heavy computational burden limits its wide-spread uses.
To address the scalability issue, we propose an implicit generative
learning-based framework called SPOT (Scalable Push-forward of Optimal
Transport). Specifically, we approximate the optimal transport plan by a
pushforward of a reference distribution, and cast the optimal transport problem
into a minimax problem. We then can solve OT problems efficiently using primal
dual stochastic gradient-type algorithms. We also show that we can recover the
density of the optimal transport plan using neural ordinary differential
equations. Numerical experiments on both synthetic and real datasets illustrate
that SPOT is robust and has favorable convergence behavior. SPOT also allows us
to efficiently sample from the optimal transport plan, which benefits
downstream applications such as domain adaptation.
