An Homogeneous Unbalanced Regularized Optimal Transport model with applications to Optimal Transport with Boundary

TL;DR

This paper introduces a modified homogeneous unbalanced regularized optimal transport (HUROT) model that preserves homogeneity properties and improves regularization in boundary transport problems, addressing limitations of standard models.

Contribution

It proposes a new HUROT model with a modified entropic regularization that maintains homogeneity, enhancing optimal transport with boundary applications.

Findings

HUROT preserves homogeneity in unbalanced OT models.

Standard UROT models may lack homogeneity with certain divergences.

HUROT improves regularization in boundary optimal transport scenarios.

Abstract

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.