动态最优输运与函数提升的联系
On the Connection between Dynamical Optimal Transport and Functional Lifting
摘要：
函数提升方法提供了一种工具，用于通过将困难的非凸问题嵌入到更大的空间中来近似解决这些问题。在这项工作中，我们研究了一个数学上严格的配方的基础上嵌入到空间的逐点概率测度在一个固定的范围$Γ$。有趣的是，这种方法可以作为动态最优输运理论的推广。施加已建立的连续性方程作为约束对应于一阶正则化的变分模型。通过修改连续性方程，该方法也可以扩展到高阶正则化模型。
英文摘要：
Functional lifting methods provide a tool for approximating solutions of difficult non-convex problems by embedding them into a larger space. In this work, we investigate a mathematically rigorous formulation based on embedding into the space of pointwise probability measures over a fixed range $Γ$. Interestingly, this approach can be derived as a generalization of the theory of dynamical optimal transport. Imposing the established continuity equation as a constraint corresponds to variational models with first-order regularization. By modifying the continuity equation, the approach can also be extended to models with higher-order regularization.
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