Ricci curvature for parametric statistics via optimal transport

Wuchen Li UCLA Guido Montufar UCLA

Differential Geometry Information Theory Machine Learning mathscidoc:2004.41001

Information geometry, 2020.1
We define the notion of a Ricci curvature lower bound for parametrized statistical models. Following the seminal ideas of Lott–Sturm–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, which are based on Fisher information matrix and Wasserstein Christoffel symbols, 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.
Optimal transport; Information geometry; Geometric analysis; Machine learning
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  title={Ricci curvature for parametric statistics via optimal transport},
  author={Wuchen Li, and Guido Montufar},
  booktitle={Information geometry},
Wuchen Li, and Guido Montufar. Ricci curvature for parametric statistics via optimal transport. 2020. In Information geometry. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20200427013101859076641.
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