On the Convergence of Semi-Relaxed Sinkhorn with Marginal Constraint and OT Distance Gaps

Abstract: This paper presents consideration of the Semi-Relaxed Sinkhorn (SR-Sinkhorn) algorithm for the semi-relaxed optimal transport (SROT) problem, which relaxes one marginal constraint of the standard OT problem. For evaluation of how the constraint relaxation affects the algorithm behavior and solution, it is vitally necessary to present the theoretical convergence analysis in terms not only of the functional value gap, but also of the marginal constraint gap as well as the OT distance gap. However, no existing work has addressed all analyses simultaneously. To this end, this paper presents a comprehensive convergence analysis for SR-Sinkhorn. After presenting the $\epsilon$-approximation of the functional value gap based on a new proof strategy and exploiting this proof strategy, we give the upper bound of the marginal constraint gap. We also provide its convergence to the $\epsilon$-approximation when two distributions are in the probability simplex. Furthermore, the convergence analysis of the OT distance gap to the $\epsilon$-approximation is given as assisted by the obtained marginal constraint gap. The latter two theoretical results are the first results presented in the literature related to the SROT problem.