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Exact optimal transport plan for Gaussian mixture models

Determine the exact Wasserstein-2 optimal transport plan π* between two Gaussian mixture model probability distributions, i.e., construct the optimal coupling that minimizes the expected squared Euclidean cost ∫|x−y|^2 dπ(x,y) subject to the marginals being the given Gaussian mixtures with specified component means, covariances, and weights, thereby providing a closed-form or fully explicit characterization of π* for Gaussian mixture models.

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Background

In the development of optimal transport flow matching (OTFM) controllers, training requires samples from the Wasserstein optimal coupling πOT between source and target distributions. For practical high-dimensional problems, the authors resort to empirical approximations (e.g., Sinkhorn) because the exact optimal coupling is often unavailable.

Even for seemingly simple pairs of distributions like Gaussian mixtures, the exact optimal transport plan is not known in closed form. This gap limits the ability to construct provably optimal controllers and to analyze dissipation precisely, motivating a concrete problem: explicit characterization of the optimal coupling for Gaussian mixture models.

References

Note that even for simple Gaussian mixture distributions, the Optimal Transport Plan is not known exactly, see for a first attempt at characterizing OT on the submanifold of Gaussian mixture models (GMM)

— Universal energy-speed-accuracy trade-offs in driven nonequilibrium systems (2402.17931 - Klinger et al., 27 Feb 2024) in Subsection “Applications to multimodal distributions and interacting particle systems,” paragraph “Bimodal optimal driving,” footnote (after Eq. (ot8))