  In optimal transport (OT), a Monge map is known as a mapping that transports
a source distribution to a target distribution in the most cost-efficient way.
Recently, multiple neural estimators for Monge maps have been developed and
applied in diverse unpaired domain translation tasks, e.g. in single-cell
biology and computer vision. However, the classic OT framework enforces mass
conservation, which makes it prone to outliers and limits its applicability in
real-world scenarios. The latter can be particularly harmful in OT domain
translation tasks, where the relative position of a sample within a
distribution is explicitly taken into account. While unbalanced OT tackles this
challenge in the discrete setting, its integration into neural Monge map
estimators has received limited attention. We propose a theoretically grounded
method to incorporate unbalancedness into any Monge map estimator. We improve
existing estimators to model cell trajectories over time and to predict
cellular responses to perturbations. Moreover, our approach seamlessly
integrates with the OT flow matching (OT-FM) framework. While we show that
OT-FM performs competitively in image translation, we further improve
performance by incorporating unbalancedness (UOT-FM), which better preserves
relevant features. We hence establish UOT-FM as a principled method for
unpaired image translation.
