The Geometry Structure And Application Of Statistical Manifolds And Matrix Manifolds

The geometric structures of statistical manifolds and matrix manifolds and their applications are investigated in this thesis in view of information geometric method. First, the geometric structures and local instabilities of two entropic dynamical mani-folds are studied. At the same time, the Riemannian means of the generalized orthogo-nal group SO(n, k) and the Poincare group P(n,1) are obtained respectively by means of the natural gradient algorithms. The optimal control of the special Euclidean sys-tem SE(n) is discussed in the thesis. Furthermore, the variations of the general linear group GL(n, R) and Riemannian manifold are investigated and the associated results obtained are similar to the ones of classical variation theorem. At last, the geometric structure and property of manifold named B-spline are considered. The whole thesis is divided into six chapters.In chapter 1, the basic theory of information geometry is introduced and then the associated backgrounds about the problems we will discuss in this thesis are briefly introduced. Finally, the main results obtained in the thesis are summarized.In chapter 2, we focus on two kinds of entropic dynamical manifolds which are built up by two mixture distributions based on three important probabilities respectively. Two kinds of Riemannian metrics on the entropic dynamical manifolds as Fisher metrics are defined from the viewpoint of information geometry. The geometric structures and the instabilities of their Jacobi fields on manifolds are investigated. Furthermore, in the second entropic dynamical manifold, another important character of chaos-Lyapunov exponent is given. It is another powerful reason for the local instability of this manifold. The associated geometric structures and properties especially for the local instabilities about submanifolds of these two kinds of manifolds are considered at the end of this chapter.In chapter 3, the Riemannian means of the generalized orthogonal group SO(n, k) and the Poincare group P(n,1) are considered in view of the information geometric method. It is well known that the generalized orthogonal group and the Poincare group play important roles in both theoretic and applied studies among the noncompact ma-trix Lie groups. The geodesies between any two points on these two matrix Lie groups are presented with the help of the left invariant metric. Then the Riemannian means defined by minimizing the summation of the geodesic distance from some point to a finite set of the given points are investigated in these two matrix groups respectively. Furthermore, some numerical simulations are shown to illustrate our outcome based on the natural gradient descent algorithm for calculating the Riemannian means on special cases of the generalized orthogonal group and the Poincare group.In chapter 4, we discuss the optimal control problem about the special Euclidean system SE(n) consisted by the special Euclidean matrixes. Considering the optimal control problem on the control system of the special Euclidean group which system output only relies on its input is meaningful for its importance in practical application. The optimal control considered here is described as the output matrix is as close as possible to the target matrix by adjusting the system input. The geodesic distance is adopted as the measure of the difference between the output and the target, then the trajectory of the control input obtained in the process that the initial output matrix evolves to the final output matrix is achieved. Furthermore, some numerical simulations are shown to illustrate our outcomes based on the natural gradient descent algorithm for optimizing the control system of the special Euclidean group.In chapter 5, we investigate variations on the general linear group GL(n,R) and Riemannian manifold respectively. Two Euler-Lagrange equations on GL(n, R) and Riemannian manifold are given so that the associated functions achieve the minimums respectively. Through the study, we can draw a conclusion that the variation principle and character are exist on GL(n,R) and Riemannian manifold. The variation of PD(n), which is a submanifold of GL(n,R), is discussed and an example about the variation of 2-order positive definite matric is given at last.In chapter 6, the geometric structure and property of B-spline manifold are s-tudied. Some geometric quantities such as Riemannian curvature vector Rijkl, Ricci curvature vector and so on are achieved and at the same time the result that B-spline manifold is ±1 flat is obtained. Finally, an example proposed illustrates the validity of the results obtained in this chapter.