This paper is on efficient implementation techniques of Elliptic Curve Cryptography. In particular, we improve timings for Jacobiquartic (3M+4S) and Hessian (7M+1S or 3M+6S) doubling operations. We provide a faster mixed-addition (7M+3S+1d) on modified Jacobiquartic coordinates. We introduce tripling formulae for Jacobi-quartic (4M+11S+2d), Jacobi-intersection (4M+10S+5d or 7M+7S+3d), Edwards (...

Linear mixed-effects models are an important class of statistical models that are not only used directly in many fields of applications but also used as iterative steps in fitting other types of mixed-effects models, such as generalized linear mixed models. The parameters in these models are typically estimated by maximum likelihood (ML) or restricted maximum likelihood (REML). In general there...

The question of how to parallelize the stochastic gradient descent (SGD) method has received much attention in the literature. In this paper, we focus instead on batch methods that use a sizeable fraction of the training set at each iteration to facilitate parallelism, and that employ second-order information. In order to improve the learning process, we follow a multi-batch approach in which t...

Several well known integral stochastic orders (like the convex order, the supermodular order, etc.) can be defined in terms of the Hessian matrix of a class of functions. Here we consider a generic Hessian order, i.e., an integral stochastic order defined through a convex coneH of Hessian matrices, and we prove that if two random vectors are ordered by the Hessian order, then their means are eq...

We consider variants of trust-region and cubic regularization methods for nonconvex optimization, in which the Hessian matrix is approximated. Under mild conditions on the inexact Hessian, and using approximate solution of the corresponding sub-problems, we provide iteration complexity to achieve ǫ-approximate second-order optimality which have shown to be tight. Our Hessian approximation condi...

We describe computational experience with automatic differentiation of mathematical programming problems expressed in the modeling language AMPL. Nonlinear expressions are translated to loop-free code, which makes it easy to compute gradients and Jacobians by backward automatic differentiation. The nonlinear expressions may be interpreted or, to gain some evaluation speed at the cost of increas...

To efficiently solve a large scale unconstrained minimization problem with a dense Hessian matrix, this paper proposes to use an incomplete Hessian matrix to define a new modified Newton method, called the incomplete Hessian Newton method (IHN). A theoretical analysis shows that IHN is convergent globally, and has a linear rate of convergence with a properly selected symmetric, positive definit...

Continuing our previous work [6, Inverse Problems, 2012, 28, 055002] and [5, Inverse Problems, 2012, 28, 055001], we address the ill-posedness of the inverse scattering problem of electromagnetic waves due to an inhomogeneous medium by studying the Hessian of the data misfit. We derive and analyze the Hessian in both Hölder and Sobolev spaces. Using an integral equation approach based on Newton...

This paper describes an implementation of the L-BFGS method designed to deal with two adversarial situations. The first occurs in distributed computing environments where some of the computational nodes devoted to the evaluation of the function and gradient are unable to return results on time. A similar challenge occurs in a multi-batch approach in which the data points used to compute functio...

This paper proposes a modification to the simultaneous per tu rba t ion stochastic approximation (SPSA) methods based on the comparisons made between the first o rder and the second order SPSA (1SPSA and 2SPSA) algori thms f rom the perspective of loss function Hessian. At finite iterations, the convergence rate depends on the matr ix conditioning of the loss function Hessian. It is shown that ...