Dealing with moment measures via entropy and optimal transport
Dealing with moment measures via entropy and optimal transport
复制标题
通过熵和最优传输处理矩量度
DOI：
10.1016/j.jfa.2016.04.009
复制
发表时间：
2015
期刊：
影响因子：
--
通讯作者：
F. Santambrogio
中科院分区：
文献类型：
--
作者：
F. Santambrogio
Cordero-Erausquin和Klartag在最近的一篇论文中给出了R d上测度μ的一个特征，它可以表示为合适的凸函数u的矩测度，即对于u：R d→ R <${+∞}具有（<$u）# e− u的形式，并通过变分方法在凸函数类中找到相应的u。在这里，我们提出了一个纯粹的基于最优传输的方法来检索相同的结果。变分问题变成了密度ρ之间的熵和运输成本的最小化，优化器ρ变成了e-u。这就需要发展一些估计和相应的泛函，这是自然的最佳运输的一些连续性结果。位移凸性的概念在极小元的刻画和唯一性中起着至关重要的作用。
A recent paper by Cordero-Erausquin and Klartag provides a characterization of the measures μ on R d which can be expressed as the moment measures of suitable convex functions u, ie are of the form (∇ u)# e− u for u: R d→ R∪{+∞} and finds the corresponding u by a variational method in the class of convex functions. Here we propose a purely optimal-transport-based method to retrieve the same result. The variational problem becomes the minimization of an entropy and a transport cost among densities ρ and the optimizer ρ turns out to be e− u. This requires to develop some estimates and some semicontinuity results for the corresponding functionals which are natural in optimal transport. The notion of displacement convexity plays a crucial role in the characterization and uniqueness of the minimizers.