Equivalence of optimal transport problems to regularization on the family of f-divergences

TL;DR

This paper proves that optimal transport problems regularized by different f-divergences can be equivalent under certain transformations, sharing the same solution, revealing a fundamental structural connection.

Contribution

It establishes a formal equivalence between OT problems regularized by different f-divergences through a transformation of the cost function.

Findings

OT problems with different f-divergence regularizations share the same solution under transformation.

Structural equivalence is demonstrated in Polish spaces with bounded cost functions.

The result unifies various regularized OT formulations under a common framework.

Abstract

This work establishes that an optimal transport~(OT) problem regularized by a given $f$-divergence admits the same solution as another OT problem regularized by a different $g$-divergence, under an appropriate transformation of the cost function. This structural equivalence between OT problems regularized by distinct divergences, in the sense of sharing the same unique minimizer, is demonstrated within the framework of Polish spaces with bounded cost functions.