SyMOT-Flow: Learning optimal transport flow for two arbitrary distributions with maximum mean discrepancy

Finding a transformation between two unknown probability distributions from samples is crucial for modeling complex data distributions and perform tasks such as density estimation, sample generation, and statistical inference. One powerful framework for such transformations is normalizing flow, which transforms an unknown distribution into a standard normal distribution using an invertible network. In this paper, we introduce a novel model called SyMOT-Flow that trains an invertible transformation by minimizing the symmetric maximum mean discrepancy between samples from two unknown distributions, and we incorporate an optimal transport cost as regularization to obtain a short-distance and interpretable transformation. The resulted transformation leads to more stable and accurate sample generation. We establish several theoretical results for the proposed model and demonstrate its effectiveness with low-dimensional illustrative examples as well as high-dimensional generative samples obtained through the forward and reverse flows.