基于最优输运的参数统计的里奇曲率
Ricci curvature for parametric statistics via optimal transport
摘要：
我们详细阐述了参数化统计模型的里奇曲率下界的概念。在洛特-斯特鲁姆-维拉尼的开创性思想之后，我们基于Wasserstein统计流形中Kullback-Leibler散度的测地凸性定义了这个概念，Wasserstein统计流形是一个具有Wasserstein度量张量结构的概率分布流形。在这些定义中，里奇曲率与信息几何和Wasserstein几何都有关。这些定义使我们能够制定的收敛速度的Wasserstein梯度流和信息功能的不等式在参数空间的界限。我们讨论了指数族模型的里奇曲率下界和收敛速度的例子。
英文摘要：
We elaborate the notion of a Ricci curvature lower bound for parametrized statistical models. Following the seminal ideas of Lott-Strum-Villani, we define this notion based on the geodesic convexity of the Kullback-Leibler divergence in a Wasserstein statistical manifold, that is, a manifold of probability distributions endowed with a Wasserstein metric tensor structure. Within these definitions, the Ricci curvature is related to both, information geometry and Wasserstein geometry. These definitions allow us to formulate bounds on the convergence rate of Wasserstein gradient flows and information functional inequalities in parameter space. We discuss examples of Ricci curvature lower bounds and convergence rates in exponential family models.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
