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.
Factor models are a class of powerful statistical models that have been widely used to deal with dependent measurements that arise frequently from various applications from genomics and neuroscience to economics and finance. As data are collected at an ever-growing scale, statistical machine learning faces some new challenges: high dimensionality, strong dependence among observed variables, heavy-tailed variables and heterogeneity. High-dimensional robust factor analysis serves as a powerful toolkit to conquer these challenges.
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