On a convergence property of a geometrical algorithm for statistical manifolds

Information geometry is a framework to analyze statistical inference and machine learning[2]. Geometrically, statistical inference and many machine learning algorithms can be regarded as procedures to find a projection to a model subspace from a given data point. In this paper, we focus on an algorithm to find the projection. Since the projection is given by minimizing a divergence, a common approach to finding the projection is a gradient-based method[6]. However, such an approach is not applicable in some cases. For instance, several attempts to extend the information geometrical framework to nonparametric cases[3, 9, 13, 15], where we need to consider a function space or each data is represented as a point process. In such a case, it is difficult to compute the derivative of divergence that is necessary for gradient-based methods, and in some cases, it is difficult to deal with the coordinate explicitly. Takano et al.[15] proposed a geometrical algorithm to find the projection for nonparametric e-mixture distribution, where the model subspace is spanned by several empirical distributions. The algorithm that is derived based on the generalized Pythagorean theorem only depends on the values of divergences.