Large-Scale Optimal Transport and Mapping Estimation
This work addresses the fundamental challenge of distribution mapping for tasks like domain adaptation and generative modeling, presenting a scalable and theoretically sound method.
The paper tackles the problem of learning an optimal map between distributions by introducing a two-step approach that first learns an optimal transport plan using a stochastic dual method, scaling better with large samples, and then estimates a Monge map via deep neural networks for generalization. It demonstrates applications in domain adaptation and generative modeling, with theoretical convergence guarantees.
This paper presents a novel two-step approach for the fundamental problem of learning an optimal map from one distribution to another. First, we learn an optimal transport (OT) plan, which can be thought as a one-to-many map between the two distributions. To that end, we propose a stochastic dual approach of regularized OT, and show empirically that it scales better than a recent related approach when the amount of samples is very large. Second, we estimate a \textit{Monge map} as a deep neural network learned by approximating the barycentric projection of the previously-obtained OT plan. This parameterization allows generalization of the mapping outside the support of the input measure. We prove two theoretical stability results of regularized OT which show that our estimations converge to the OT plan and Monge map between the underlying continuous measures. We showcase our proposed approach on two applications: domain adaptation and generative modeling.