The e-PCA and m-PCA: dimension reduction of parameters by information geometry

We propose a method for extracting a low dimensional structure from a set of parameters of probability distributions. By an information geometrical interpretation, we show that there exist two kinds of possible flat structures for fitting (e-PCA and m-PCA). We derive alternating procedures to find the low dimensional structures. Each alternating procedure can be written in a nonlinear equation. It can be solved analytically in some special cases. Otherwise, we need to apply gradient type methods that we also derive. Since the overall algorithm may converge to a local optimum, we propose a method to find a good initial solution by using metric information.