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 $L^{(\alpha)}$-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 $\mathcal{F}^{(\alpha)}$-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 $L^{(\alpha)}$-PCA problem which generalizes the exponential family PCA.

翻译：在信息几何学中,近年来正在积极调查通用指数式家庭以及具有曲线的统计元体。在本文中,我们考虑了由对数 $ {( ALpha)}} $- divegence 所引发的统计元数,它概括了布雷格曼值的差异。众所周知,这种元数的投影成双平,与不变的负部位曲线有关,并且与第二个作者所引入的普遍指数式家庭$ mathcal{F}( ALpha)} $- family密切相关。我们的主要结果形成了统计元体的双重变形,即由原始值和双自动parolel子元组成的正方形分解变形。这种分形可以自然地用原始值和双向预测来解释,扩大了Amari所研究的双平面多元值的双倍叶化。作为应用,我们提出了一个新的 $ ⁇ (\\ dalpha)- PCA 问题,它使指数型家庭的五氯苯具有共性。