2021/05/08 by Zhixu Tao, Tao, Zhixu, Ting‐Kam Leonard Wong +1
Computer Science · Environmental Science · Physics and Astronomy · #Differential Geometry (math.DG) #FOS: Computer and information sciences #FOS: Mathematics #Geochemistry and Geologic Mapping #Methodology (stat.ME) #Soil Geostatistics and Mapping #Statistical Mechanics and Entropy
paper · pdf · doi:10.48550/arxiv.2105.07767
openalex publication_date 2021/05/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In information geometry, generalized exponential families and statistical manifolds with curvature are under active investigation in recent years. In this paper we consider the statistical manifold induced by a logarithmic L(α)-divergence which generalizes the Bregman divergence. It is known that such a manifold is dually projectively flat with constant negative sectional curvature, and is closely related to the F(α)-family, a generalized exponential family introduced by the second author. Our main result constructs a dual foliation of the statistical manifold, i.e., an orthogonal decomposition consisting of primal and dual autoparallel submanifolds. This decomposition, which can be naturally interpreted in terms of primal and dual projections with respect to the logarithmic divergence, extends the dual foliation of a dually flat manifold studied by Amari. As an application, we formulate a new L(α)-PCA problem which generalizes the exponential family PCA.