Entropic Optimal Transport between Unbalanced Gaussian Measures has a Closed Form

Very few optimal transport (OT) problems admit closed form solutions. Among those, OT on the space of Gaussians induces the Bures metric between covariance matrices (also known as Fr\'echet distance). In this paper, we extend this celebrated result by providing a closed form expression of the entropic regularization of OT between Gaussians. Entropic regularization is extensively used in practice as it provides algorithmic solutions in cases that cannot be guided by any known ``ground truth''. Contrary to the Wassertein-Bures unregularized case distance, the closed form we obtain is differentiable everywhere, even for Gaussians with degenerate covariance matrices, which allows to safely use Wasserstein-Bures distances in gradient-based optimization scenarios. We obtain this closed form solution by solving the fixed-point equation behind Sinkhorn's algorithm, the default method for solving entropic regularized OT. Remarkably, this approach extends to the generalized unbalanced case --- where Gaussian measures are scaled by positive constants. This extension leads to a closed form expression for unbalanced Gaussians as well, and highlights the mass transportation / destruction trade-off seen in unbalanced optimal transport. Moreover, in both settings, we show that the optimal transportation plans are (scaled) Gaussians and provide analytical formulae of their parameters. These formulae constitute the first non-trivial closed forms for entropy-regularized optimal transport, thus providing a ground truth for the analysis of entropic OT and Sinkhorn's algorithm.

• Algorithms and Theory

• Machine Intelligence