Simultaneous Neural Optimal Transport

Optimal Transport (OT) provides a principled framework for learning transformations between probability distributions from unpaired samples. In many applications, however, a single transformation must map several source distributions to a common target distribution. For example, image restoration may require handling different types of degradation without knowing the degradation of each input at inference time. Simply pooling the source distributions only encourages alignment with the target at the aggregate level and may leave individual sources misaligned. In this paper, we consider the simultaneous OT problem, which seeks a shared transport map that minimizes the average transport cost while aligning each source distribution with a prescribed target. We derive a max-min formulation of an unbalanced extension and propose SimNOT, a neural method based on this formulation. We illustrate its application to image restoration, where a single model handles multiple degradation types using a common collection of clean target images.