揭示半对偶最优传输中映射收敛的条件，解释算法迭代不均衡现象。
Stability of the Monge Map in Semi-Dual Optimal Transport
- 通过分析半对偶结构，提出映射收敛的充要条件
- 发现无需对偶势能最优即可保证映射收敛
- 解释数值算法中映射更新比势能更耗迭代的原因
本文揭示了最优传输问题半对偶形式具有退化的鞍点结构，其数值求解等价于一个约束优化问题。我们推导出在不依赖对偶势能最优性的前提下，Monge映射收敛的必要与充分条件。该分析有助于解释实际应用中，数值算法通常需要更多迭代次数来更新传输映射而非对偶势能的现象。
原文摘要 · Abstract (English)
This paper shows that the semi-dual formulation of the optimal transport problem has a degenerate saddle-point structure, and that its numerical solution is equivalent to solving a constrained optimization problem. We derive necessary and sufficient conditions for the convergence of Monge maps without requiring optimality of the dual potential. This analysis helps explain why, in practice, numerical algorithms often require more iterations to update the transport map than the potential.
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