Information Geometry for Random Walk

The random walk（RW）is investigated from the viewpoint of information geometry and shown to be an exponential family distribution.It has a dual coordinate system and a dual geometric structure.Then submanifolds of RW manifold is studied,and the e-flat hierarchical structure and the orthogonal foliations of RW manifold are obtained.Finally,using the Kullback-Leibler divergence,the projections are given from the RW manifold to its submanifolds.

Information Geometry and Low Density Parity Check Codes

The application of information geometry in the low density parity check（LDPC）codes based on the work of Ikeda and Amari is considered.The method is to turn the decoding process into the change of the parameter.When the LDPC decoding procedure converges,both the convergent probability distribution and the true probability distribution belong to the same submanifold,but this does not mean they are equimarginal,that points out the origin of the decoding error.

Information Geometry of Positive Definite Matrices

A new Riemannian metric for positive definite matrices is defined and its geometric structures are investigated by means of dual connections introduced to statistical analysis by S. Amari. A few interesting results are obtained and some of those obtained by other authors are extended in our research.

Information Geometry of GARCH Model

A statistical manifold of non-exponential type coming from a model for economics describing stock return process is constructed, with its geometric structure investigated and both Gaussian curvatures and mean curvatures of its curved exponential submanifolds deducted. A few graphs describing relevant scalar curvature, mean curvature and Gaussian curvature are also introduced.

Encoding Energy-Density as Geometry

Physicists possess an intuitive awareness of Euclidian space and time and Galilean transformation, and are then challenged with Minkowski space-time and Einstein’s curved space-time. Relativistic experiments support the “time-dilation” interpretation and others support “curved space-time” interpretation. In this, and related work, we investigate the key issues in terms of the intuitive space-time frame. In particular, we provide alternative approaches to explain “time dilation” and to explain the energy density for gravity systems. We approach the latter problem from an information perspective.

Information Geometric Structures for the Thermodynamic Manifold

In view of information geometry,the state space S of thermodynamic parameters is investigated.First a Riemannian metric for S is defined and then the α-geometric structures of S is given.Some of results obtained by other authors are extended.

Solution of Algebraic Lyapunov Equation on Positive-Definite Hermitian Matrices by Using Extended Hamiltonian Algorithm

This communique is opted to study the approximate solution of the Algebraic Lyapunov equation on the manifold of positive-definite Hermitian matrices.We choose the geodesic distance between􀀀AHX􀀀XA and P as the cost function,and put forward the Extended Hamiltonian algorithm(EHA)and Natural gradient algorithm(NGA)for the solution.Finally,several numerical experiments give you an idea about the effectiveness of the proposed algorithms.We also show the comparison between these two algorithms EHA and NGA.Obtained results are provided and analyzed graphically.We also conclude that the extended Hamiltonian algorithm has better convergence speed than the natural gradient algorithm,whereas the trajectory of the solution matrix is optimal in case of Natural gradient algorithm(NGA)as compared to Extended Hamiltonian Algorithm(EHA).The aim of this paper is to show that the Extended Hamiltonian algorithm(EHA)has superior convergence properties as compared to Natural gradient algorithm(NGA).Upto the best of author’s knowledge,no approximate solution of the Algebraic Lyapunov equation on the manifold of positive-definite Hermitian matrices is found so far in the literature.

Information geometry in optimization,machine learning and statistical inference

The present article gives an introduction to information geometry and surveys its applications in the area of machine learning,optimization and statistical inference.Information geometry is explained intuitively by using divergence functions introduced in a manifold of probability distributions and other general manifolds.They give a Riemannian structure together with a pair of dual flatness criteria.Many manifolds are dually flat.When a manifold is dually flat,a generalized Pythagorean theorem and related projection theorem are introduced.They provide useful means for various approximation and optimization problems.We apply them to alternative minimization problems,Ying-Yang machines and belief propagation algorithm in machine learning.

A Re-Interpretation of Quasi-Local Mass

Because the equivalence principle forbids local mass density, we cannot formulate general relativistic mass as an integral over mass density as in Newtonian gravity. This century-old problem was addressed forty years ago by Penrose, and many papers have since extended the concept. Currently there is no satisfactory physical understanding of the nature of quasi-local mass. In this paper I review the key issues, the current status, and propose an alternative interpretation of the problem of local mass and energy density for gravity systems from an information perspective.

Geometrical Structures of Certain Class of Statistical Manifolds

The geometrical structures of the certain class of statistical manifolds are investigated. The geometwhich includes the original geometrical metrics of S.Amari.

Information geometry in neural spike sequences

An information geometrical method is developed for characterizing or classifying neurons in cortical areas whose spike rates fluctuate in time.The interspike intervals(ISIs)of a spike sequence of a neuron is modeled as a gamma process with a time-variant spike rate,a fixed shape parameter and a fixed absolute refractory period.We formulate the problem of estimating the fixed parameters as semiparametric estimation and apply an information geometrical method to derive the optimal estimators from a statistical viewpoint.

The α-Geometric Structures on Manifold of Positive Definite Hermite Matrices

Geometric structures of a manifold of positive definite Hermite matrices are considered from the viewpoint of information geometry.A Riemannian metric is defined and dual α-connections are introduced.Then the fact that the manifold is ±l-flat is shown.Moreover,the divergence of two points on the manifold is given through dual potential functions.Furthermore,the optimal approximation of a point onto the submanifold is gotten.Finally,some simulations are given to illustrate our results.