Neural Optimal Transport with General Cost Functionals
This addresses the challenge of applying optimal transport with custom costs in machine learning, enabling tasks like distribution mapping with class preservation, though it is incremental by extending existing methods to continuous settings.
The paper tackles the problem of computing optimal transport plans for general cost functionals, which offer more flexibility than standard Euclidean costs, by introducing a neural network-based algorithm that provides continuous out-of-sample estimation and theoretical error analysis, achieving competitive performance in high-dimensional spaces like images.
We introduce a novel neural network-based algorithm to compute optimal transport (OT) plans for general cost functionals. In contrast to common Euclidean costs, i.e., $\ell^1$ or $\ell^2$, such functionals provide more flexibility and allow using auxiliary information, such as class labels, to construct the required transport map. Existing methods for general costs are discrete and have limitations in practice, i.e. they do not provide an out-of-sample estimation. We address the challenge of designing a continuous OT approach for general costs that generalizes to new data points in high-dimensional spaces, such as images. Additionally, we provide the theoretical error analysis for our recovered transport plans. As an application, we construct a cost functional to map data distributions while preserving the class-wise structure.