Hessian-free (HF) optimization has been successfully used for training deep autoencoders and recurrent networks. HF uses the conjugate gradient algorithm to construct update directions through curvature-vector products that can be computed on the same order of time as gradients. In this paper we exploit this property and study stochastic HF with gradient and curvature mini-batches independent o...

Chen’s first inequality for statistical submanifolds in Hessian manifolds of constant curvature was obtained by B.-Y. Chen et al. Other particular cases inequalities a setting were given different authors. The objective the present article is to establish general curvature.

The geometry of Hessian manifolds is a fruitful branch physics, statistics, Kaehlerian and affine differential geometry. study inequalities for statistical submanifolds in constant curvature was truly initiated 2018 by Mihai, A. I. who dealt with Chen-Ricci Euler inequalities. Later on, Siddiqui, A.N., Ahmad K. Ozel C. came the Casorati inequality same ambient space using algebraic technique. A...

In this paper, we generalize the Hessian comparison theorems and Laplacian comparison theorems described in [16, 18], then give some applications under various curvature conditions.

Quasi-Newton methods are reliable and efficient on a wide range of problems, but they can require many iterations if the problem is ill-conditioned or if a poor initial estimate of the Hessian is used. In this paper, we discuss methods designed to be more efficient in these situations. All the methods to be considered exploit the fact that quasi-Newton methods accumulate approximate second-deri...

We present a new matrix-free method for the computation of the negative curvature direction in large scale unconstrained problems. We describe a curvilinear method which uses a combination of a quasi-Newton direction and a negative curvature direction. We propose an algorithm for the computation of the search directions which uses information of two specific L-BFGS matrices in such a way that a...

Hessian information speeds convergence substantially in motion optimization. The better the Hessian approximation the better the convergence. But how good is a given approximation theoretically? How much are we losing? This paper addresses that question and proves that for a particularly popular and empirically strong approximation known as the Gauss-Newton approximation, we actually lose very ...

In this work we develop Curvature Propagation (CP), a general technique for efficiently computing unbiased approximations of the Hessian of any function that is computed using a computational graph. At the cost of roughly two gradient evaluations, CP can give a rank-1 approximation of the whole Hessian, and can be repeatedly applied to give increasingly precise unbiased estimates of any or all ...

In this paper, we establish the curvature estimates for a class of Hessian type equations. Some applications are also discussed.

In this paper, inspired by P.M.H. Wilson’s paper on sectional curvatures of Kähler moduli [31], we concentrate on the case where f is a homogeneous polynomial (also called a “form”) of degree d at least 2. Following Okonek and van de Ven [23], Wilson considers the “index cone,” the open subset where the Hessian matrix of f is Lorentzian (that is, of signature (1, ∗)) and f is positive. He restr...