A Preliminaries on the Sinkhorn Potentials Lemma A.1 (Lemma A.2 elaborated)

One can observe that the Sinkhorn potentials are not unique. Under Assumption A.2, for a fixed pair of Assume Assumptions A.2 and A.3, and denote In this section, we provide several lemmas to show the Lipschitz continuity (w.r.t. the underlying These lemmas will be used in the convergence analysis and the mean field analysis for SD . Please see the proof in Appendix C.3. It would be an interesting future work to remove this factor . W e note that the Lemma B.1 is strictly stronger than preexisting results: (1) Proposition 13 of Feydy et al. [2019] only shows that the dual potentials are continuous (not Lipschitz continuous) This is strictly weaker than (i) of Lemma B.1.