The dually flat structure introduced by Amari–Nagaoka is highlighted in information geometry and related fields. In practical applications, however, the underlying pseudo-Riemannian metric may often be degenerate, such an excellent geometric rarely defined on entire space. To fix this trouble, present paper, we propose a novel generalization of for certain class singular models from viewpoint L...

Classical information geometry prescribes, on the parametric family of probability functions Mθ : (i) a Riemannian metric given by the Fisher information; (ii) a pair of dual connections (giving rise to the family of α-connections) that preserve the metric under parallel transport by their joint actions; and (iii) a family of (non-symmetric) divergence functions (α-divergence) defined on Mθ ×Mθ...

The Gaussian scale space for an n-dimensional image L(x) is defined as its n + 1 dimensional extension L(x, t) being the solution of ∂tL = ∆L,Lt=0 = L(x). A hierarchical structure in L(x, t) is derived by combining the critical curves (∇xL(x, t) = 0) and special points on them, viz. catastrophe points (where detH = 0, H being the Hessian matrix) and scale space saddles (where ∆L = 0), with iso-...

We classify the paracontact Riemannian manifolds that their Riemannian curvature satisfies in the certain condition and we show that this classification is hold for the special cases semi-symmetric and locally symmetric spaces. Finally we study paracontact Riemannian manifolds satisfying R(X, ξ).S = 0, where S is the Ricci tensor.

The k-Hessian is the k-trace, or the kth elementary symmetric polynomial of eigenvalues of the Hessian matrix. When k ≥ 2, the k-Hessian equation is a fully nonlinear partial differential equations. It is elliptic when restricted to k-admissible functions. In this paper we establish the existence and regularity of k-admissible solutions to the Dirichlet problem of the k-Hessian equation. By a g...

We present a new Hessian estimator based on the simultaneous perturbation procedure, that requires three system simulations regardless of the parameter dimension. We then present two Newton-based simulation optimization algorithms that incorporate this Hessian estimator. The two algorithms differ primarily in the manner in which the Hessian estimate is used. Both our algorithms do not compute t...

We investigate the computation of Hessian matrices via Automatic Differentiation, using a graph model and an algebraic model. The graph model reveals the inherent symmetries involved in calculating the Hessian. The algebraic model, based on Griewank and Walther’s state transformations [7], synthesizes the calculation of the Hessian as a formula. These dual points of view, graphical and algebrai...

Limited-memory BFGS quasi-Newton methods approximate the Hessian matrix of second derivatives by the sum of a diagonal matrix and a fixed number of rank-one matrices. These methods are particularly effective for large problems in which the approximate Hessian cannot be stored explicitly. It can be shown that the conventional BFGS method accumulates approximate curvature in a sequence of expandi...