论文标题
Schrödinger电位的梯度估计：收敛到Brenier图和定量稳定性
Gradient estimates for the Schrödinger potentials: convergence to the Brenier map and quantitative stability
论文作者
论文摘要
我们显示了Schrödinger电位的梯度在边际上的一般假设下以小型限制在小型限制下的收敛性，这允许无限的密度和支撑。此外，我们为Schrödinger问题（SP）的最佳值和最佳耦合提供了新的定量稳定性估计，我们以负顺序加权同质性Sobolev Norm表示。后者编码边缘之间的2-wasserstein距离的线性化行为。两种结果的证据首次突出显示了梯度界限与施罗丁电位的相关性，我们在这里完全建立了Schrödinger桥的短期行为。最后，我们讨论我们的结果如何转化为二次熵最佳传输的框架，这是SP的一种更适合于机器学习和数据科学应用的SP版本。
We show convergence of the gradients of the Schrödinger potentials to the Brenier map in the small-time limit under general assumptions on the marginals, which allow for unbounded densities and supports. Furthermore, we provide novel quantitative stability estimates for the optimal values and optimal couplings for the Schrödinger problem (SP), that we express in terms of a negative order weighted homogeneous Sobolev norm. The latter encodes the linearized behavior of the 2-Wasserstein distance between the marginals. The proofs of both results highlight for the first time the relevance of gradient bounds for Schrödinger potentials, that we establish here in full generality, in the analysis of the short-time behavior of Schrödinger bridges. Finally, we discuss how our results translate into the framework of quadratic Entropic Optimal Transport, that is a version of SP more suitable for applications in machine learning and data science.
