论文标题
熵最佳运输的位移平稳性
Displacement smoothness of entropic optimal transport
论文作者
论文摘要
将概率度量家族映射到双熵最佳运输问题解决方案的函数称为Schrödinger图。我们证明，当成本函数为$ \ MATHCAL {C}^{k+1} $带有$ k \ in \ mathbb {n}^*$的$ k \ $，则此映射是Lipschitz从$ l^2 $ - wasserstein空间连续到$ \ mathcal {C}^k $功能。我们的结果保留在紧凑的域并覆盖多核心情况。我们还在负面的sobolev指标下，与瓦斯坦（Wasserstein）相比，在成本更平稳的假设下，我们还包括规律性结果。作为应用，我们证明了熵最佳运输成本的位移平稳性以及某些涉及该功能的某些Wasserstein梯度流的适当性，包括sindhorn差异和多物种系统。
The function that maps a family of probability measures to the solution of the dual entropic optimal transport problem is known as the Schrödinger map. We prove that when the cost function is $\mathcal{C}^{k+1}$ with $k\in \mathbb{N}^*$ then this map is Lipschitz continuous from the $L^2$-Wasserstein space to the space of $\mathcal{C}^k$ functions. Our result holds on compact domains and covers the multi-marginal case. We also include regularity results under negative Sobolev metrics weaker than Wasserstein under stronger smoothness assumptions on the cost. As applications, we prove displacement smoothness of the entropic optimal transport cost and the well-posedness of certain Wasserstein gradient flows involving this functional, including the Sinkhorn divergence and a multi-species system.
