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This paper presents a globally optimal solution for unbalanced optimal transport (UOT) between Gaussian distributions by leveraging a control-theoretic dynamical extension called unbalanced density control (UDC). They demonstrate that both the static UOT problem with KL penalties and the dynamic UDC problem with linear systems admit exact Gaussian reductions, transforming infinite-dimensional variational problems into finite-dimensional optimizations. The resulting formulations, combined with covariance-steering lifting, enable SDP-based optimization for fixed mass and closed-form mass updates, offering a computationally efficient and globally optimal approach.
Forget iterative approximations – this work delivers globally optimal solutions for unbalanced optimal transport between Gaussians via a clever reduction to finite-dimensional optimization.
In this article, we study unbalanced optimal transport (UOT) and establish a control-theoretic dynamical extension, which we call the unbalanced density control (UDC), for a class of Gaussian reference measures. In the static setting, we consider UOT with quadratic transport cost and Kullback--Leibler penalties on the marginals relative to prescribed Gaussian measures. We show that the infinite-dimensional variational problem admits an exact Gaussian reduction, yielding a finite-dimensional optimization over masses, means, and covariances, together with a closed-form expression for the optimal transported mass. We then formulate UDC for discrete-time linear systems, where the initial and terminal state measures are imposed softly through KL penalties and the intermediate evolution is governed by controlled linear dynamics with quadratic control cost. For this problem, we prove that any feasible solution can be replaced, without loss of optimality, by a Gaussian initial measure and an affine-Gaussian control policy. This leads to an exact finite-dimensional reformulation and, after a standard covariance-steering lifting, to an SDP-based optimization for fixed mass, again coupled with a closed-form mass update. We further establish existence of optimal solutions and identify a sufficient condition under which the affine-Gaussian UDC policy is deterministic. These results provide globally optimal solution methods for both Gaussian UOT and Gaussian UDC. Finally, we illustrate our results with several numerical examples.
