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Entropic optimal transport: convergence of potentials

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Abstract

We study the potential functions that determine the optimal density for \(\varepsilon \)-entropically regularized optimal transport, the so-called Schrödinger potentials, and their convergence to the counterparts in classical optimal transport, the Kantorovich potentials. In the limit \(\varepsilon \rightarrow 0\) of vanishing regularization, strong compactness holds in \(L^{1}\) and cluster points are Kantorovich potentials. In particular, the Schrödinger potentials converge in \(L^{1}\) to the Kantorovich potentials as soon as the latter are unique. These results are proved for all continuous, integrable cost functions on Polish spaces. In the language of Schrödinger bridges, the limit corresponds to the small-noise regime.

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Notes

  1. We mention that [19] uses the term Schrödinger potentials for \(f_{\varepsilon }/\varepsilon ,g_{\varepsilon }/\varepsilon \) in the Schrödinger bridge context, as is natural when no parameter \(\varepsilon \) is present. On the other hand, calling \(f_{\varepsilon },g_{\varepsilon }\) potentials is more convenient in our setting, well motivated by the connection with Kantorovich potentials in Theorem 1.1, and consistent with the terminology in [17].

  2. I.e., \(I^{*}\) is the (a.s. unique) measurable function satisfying \(I^{*}\ge I_{\lambda }\) a.s. for all \(\lambda \) and \(I^{*}\le J\) a.s. for any J satisfying \(J\ge I_{\lambda }\) a.s. for all \(\lambda \). In other words, \(I^{*}\) is the supremum in the lattice of measurable functions equipped with the a.s. order.

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Authors and Affiliations

  1. Departments of Statistics and Mathematics, Columbia University, 1255 Amsterdam Avenue, New York, NY, 10027, USA

    Marcel Nutz

  2. Department of Statistics, Columbia University, 1255 Amsterdam Avenue, New York, NY, 10027, USA

    Johannes Wiesel

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Correspondence to Marcel Nutz.

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The authors thank Guillaume Carlier, Giovanni Conforti, Soumik Pal and Luca Tamanini for helpful discussions.

MN acknowledges support by an Alfred P. Sloan Fellowship and NSF Grants DMS-1812661, DMS-2106056.

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Nutz, M., Wiesel, J. Entropic optimal transport: convergence of potentials. Probab. Theory Relat. Fields 184, 401–424 (2022). https://doi.org/10.1007/s00440-021-01096-8

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