Closed Forms for Gaussian Kullback--Leibler Unbalanced Optimal Transport without Coupling Entropy

TL;DR

This paper derives explicit solutions for Gaussian unbalanced optimal transport problems with KL divergence, revealing a scaled Wasserstein coupling supported on an affine graph and providing new insights into the structure of optimal plans.

Contribution

It provides the first explicit solution for Gaussian KL unbalanced optimal transport without coupling entropy, including the covariance map and dual certificates.

Findings

The optimal plan is a scaled Wasserstein coupling supported on an affine graph.

The covariance map solves a Riccati equation with a principal-square-root representation.

Large relaxation limit recovers Gaussian Wasserstein cost for equal masses.

Abstract

We obtain an explicit solution for the static Kullback--Leibler (KL) unbalanced optimal transport problem between finite non-degenerate Gaussian measures with quadratic cost, two independent positive marginal relaxation parameters, and no entropy penalty on the coupling. The minimizer is a scaled Wasserstein coupling between two adjusted Gaussian marginals and is supported on an affine graph; in entropic Gaussian unbalanced transport, by contrast, the optimal plan is non-degenerate on the product space. The covariance map is the unique positive definite solution of a Riccati equation and admits a principal-square-root representation. Compared with the known equal-penalty Gaussian Hellinger--Kantorovich endpoint, the result treats the asymmetric two-sided Kullback--Leibler relaxation and gives the modified marginals, joint minimizer, value, and a direct quadratic KL-dual certificate. The large-relaxation limit recovers the Gaussian Wasserstein cost for equal masses.