Some Notes On Riemannian Geometry In Information Geometry

Information geometry is a subject that studies the problems in the field of information by means of Riemann method.In the process of solving nonlinear problems,traditional methods blindly linearize the nonlinear problems,but can't get satisfactory accuracy,so the method of using geometric means to deal with nonlinear problems arises at the historic moment.Moreover,information geometry is widely used in neural networks,statistical inference,statistical physics,machine learning,manifold learning and other emerging fields.From its development process,information geometry has experienced the development from classical information geometry to matrix information geometry.However,as a newly developed interdisciplinary subject,it is not known to many mathematical workers.This paper aims to introduce some important frameworks of classical information geometry and the author's reading notes,so that readers can get a glimpse of the outline of information geometry.In classical information geometry,we regard the probability density function as a statistical manifold.On this manifold,every probability density function is regarded as a point.The Riemann metric on statistical manifold is given by Fisher information matrix.Therefore,the statistical manifold becomes a Riemann manifold.It is very convenient to study the statistical manifold by introducing ?—connection as its derivative.Since ?—connection and—?—connection are dual connections,?—connection is torsion-free.Exponential distribution family and mixed distribution family are two important distribution families.?—connection plays an important role in the calculation of geometric quantities of exponential distribution family.When ?=±1,the family manifold of exponential distribution is flat.Kullbac—Leiblier divergence can be introduced to measure the distance between two points on a±1 flat manifold.The divergence only satisfies nonnegativity,but does not satisfy trigonometric inequality and symmetry.In addition,this paper also introduces some applications of information geometry,such as natural gradient algorithm and entropy dynamic model.Compared with standard gradient algorithm,natural gradient algorithm is Fisher-efficient which guarantee the higher accuracy.Information geometry has great application in entropy dynamic model.The stability of entropy dynamic model can be explained by the stability of Jacobi field corresponding to entropy dynamic model.In this paper,the mode length of Jacobi field is estimated by comparison theorem,and the linear growth of the model is obtained,which avoids complicated calculation.