  This work studies how the introduction of the entropic regularization term in
unbalanced Optimal Transport (OT) models may alter their homogeneity with
respect to the input measures. We observe that in common settings (including
balanced OT and unbalanced OT with Kullback-Leibler divergence to the
marginals), although the optimal transport cost itself is not homogeneous,
optimal transport plans and the so-called Sinkhorn divergences are indeed
homogeneous. However, homogeneity does not hold in more general Unbalanced
Regularized Optimal Transport (UROT) models, for instance those using the Total
Variation as divergence to the marginals. We propose to modify the entropic
regularization term to retrieve an UROT model that is homogeneous while
preserving most properties of the standard UROT model. We showcase the
importance of using our Homogeneous UROT (HUROT) model when it comes to
regularize Optimal Transport with Boundary, a transportation model involving a
spatially varying divergence to the marginals for which the standard
(inhomogeneous) UROT model would yield inappropriate behavior.
