论文标题
关于半积极的sindhorn的收敛，具有边际约束和OT距离差距
On the Convergence of Semi-Relaxed Sinkhorn with Marginal Constraint and OT Distance Gaps
论文作者
论文摘要
本文介绍了半积极的最佳运输（SROT）问题的半积分sindhorn（SR-Sinkhorn）算法，这使标准OT问题的边缘约束放松了。为了评估约束弛豫如何影响算法的行为和解决方案，不仅以功能值差距的术语，而且是边缘约束差距以及OT距离差距，至关重要地呈现理论收敛分析。但是，没有现有的工作同时解决所有分析。为此，本文为SR-Sinkhorn提供了全面的收敛分析。在提出了基于新的证明策略并利用此证明策略的功能价值差距的$ε$ - $ approximation之后，我们给出了边际约束差距的上限。当概率中有两个分布时，我们还将其收敛到$ε$ - approximation。此外，对OT距离差距到$ε$ - APPROXIMATION的收敛分析是由获得的边缘约束缝隙的辅助。后两个理论结果是与Srot问题有关的文献中提出的第一个结果。
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 $ε$-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 $ε$-approximation when two distributions are in the probability simplex. Furthermore, the convergence analysis of the OT distance gap to the $ε$-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.
