Unbalanced and Light Optimal Transport

While the field of continuous Entropic Optimal Transport (EOT) has been actively developing in recent years, it became evident that the classic EOT problem is prone to different issues like the sensitivity to outliers and imbalance of classes in the source and target measures. This fact inspired the development of solvers which deal with the unbalanced EOT (UEOT) problem - the generalization of EOT allowing for mitigating the mentioned issues by relaxing the marginal constraints. Surprisingly, it turns out that the existing solvers are either based on heuristic principles or heavy-weighted with complex optimization objectives involving several neural networks. We address this challenge and propose a novel theoretically-justified and lightweight unbalanced EOT solver. Our advancement consists in developing a novel view on the optimization of the UEOT problem yielding tractable and non-minimax optimization objective. We show that combined with a light parametrization recently proposed in the field our objective leads to fast, simple and effective solver. It allows solving the continuous UEOT problem in minutes on CPU. We provide illustrative examples of the performance of our solver.