An Homogeneous Unbalanced Regularized Optimal Transport model with applications to Optimal Transport with Boundary
This work addresses a theoretical limitation in optimal transport models for researchers in computational mathematics and machine learning, offering an incremental improvement for specific applications.
The authors tackled the issue of homogeneity loss in unbalanced regularized optimal transport (UROT) models due to entropic regularization, proposing a modified homogeneous UROT (HUROT) model that preserves key properties and is shown to be crucial for applications like optimal transport with boundary, where standard models fail.
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.