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Entropic Optimal Transport between Unbalanced Gaussian Measures has a Closed Form
Hicham Janati · Boris Muzellec · Gabriel Peyré · Marco Cuturi

Wed Dec 09 09:00 AM -- 11:00 AM (PST) @ Poster Session 3 #898

Although optimal transport (OT) problems admit closed form solutions in a very few notable cases, e.g. in 1D or between Gaussians, these closed forms have proved extremely fecund for practitioners to define tools inspired from the OT geometry. On the other hand, the numerical resolution of OT problems using entropic regularization has given rise to many applications, but because there are no known closed-form solutions for entropic regularized OT problems, these approaches are mostly algorithmic, not informed by elegant closed forms. In this paper, we propose to fill the void at the intersection between these two schools of thought in OT by proving that the entropy-regularized optimal transport problem between two Gaussian measures admits a closed form. Contrary to the unregularized case, for which the explicit form is given by the Wasserstein-Bures distance, the closed form we obtain is differentiable everywhere, even for Gaussians with degenerate covariance matrices. We obtain this closed form solution by solving the fixed-point equation behind Sinkhorn's algorithm, the default method for computing 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 formulas of their parameters. These formulas 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.

Author Information

Hicham Janati (Inria / ENSAE)
Boris Muzellec (Inria Paris)
Gabriel Peyré (CNRS and ENS)
Marco Cuturi (Google Brain & CREST - ENSAE)

Marco Cuturi is a research scientist at Apple, in Paris. He received his Ph.D. in 11/2005 from the Ecole des Mines de Paris in applied mathematics. Before that he graduated from National School of Statistics (ENSAE) with a master degree (MVA) from ENS Cachan. He worked as a post-doctoral researcher at the Institute of Statistical Mathematics, Tokyo, between 11/2005 and 3/2007 and then in the financial industry between 4/2007 and 9/2008. After working at the ORFE department of Princeton University as a lecturer between 2/2009 and 8/2010, he was at the Graduate School of Informatics of Kyoto University between 9/2010 and 9/2016 as a tenured associate professor. He joined ENSAE in 9/2016 as a professor, where he is now working part-time. He was at Google between 10/2018 and 1/2022. His main employment is now with Apple, since 1/2022, as a research scientist working on fundamental aspects of machine learning.

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