  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.
