Entropic optimal transport: convergence of potentials

We study the potential functions that determine the optimal density for epsilon-entropically regularized optimal transport, the so-called Schrodinger potentials, and their convergence to the counterparts in classical optimal transport, the Kantorovich potentials. In the limit epsilon -> 0 of vanishing regularization, strong compactness holds in L-1 and cluster points are Kantorovich potentials. In particular, the Schrodinger 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 Schrodinger bridges, the limit corresponds to the small-noise regime.

Automated summary

The study focuses on the potential functions determining the optimal density for epsilon-entropically regularized optimal transport, known as Schrodinger potentials, and their convergence to Kantorovich potentials in classical optimal transport. As epsilon approaches 0, strong compactness holds in L-1 and the Schrodinger potentials converge to Kantorovich potentials, particularly when the latter are unique. These results apply to all continuous, integrable cost functions on Polish spaces and the limit corresponds to the small-noise regime in the language of Schrodinger bridges.