Many methods for solving minimization problems are variants of Newton method, which requires the specification of the Hessian matrix of second derivatives. QuasiNewton methods are intended for the situation where the Hessian is expensive or difficult to calculate. Quasi-Newton methods use only first derivatives to build an approximate Hessian over a number of iterations. This approximation is u...

Since the data sizes in fully 3D PET imaging are very large, iterative image reconstruction algorithms must converge in very few iterations to be useful. One can improve the convergence rate of the conjugate-gradient (CG) algorithm by incorporating precondi-tioning operators that approximate the inverse of the Hessian of the objective function. If the 3D cylindrical PET geometry were not trunca...

In this paper we are interested in the higher-order derivatives of functions of the eigenvalues of symmetric matrices with respect to the matrix argument. We describe the formula for the k-th derivative of such functions in two general cases. The first case concerns the derivatives of the composition of an arbitrary (not necessarily symmetric) k-times differentiable function with the eigenvalue...

(3) because ‖(xa,va)‖2 ≤ L and ηt only has a finite variance. For the first term on the right-hand side in Eq (2), if the regularization parameter λ1 is sufficiently large, the Hessian matrix of the loss function specified in the paper is positive definite at the optimizer based on the property of alternating least square (Uschmajew 2012). The estimation of Θ and va is thus locally q-linearly c...

The algorithm described in this article is based on the OBS algorithm by Hassibi, Stork and Wolff ([1] and [2]). The main disadvantage of OBS is its high complexity. OBS needs to calculate the inverse Hessian to delete only one weight (thus needing much time to prune a big net) . A better algorithm should use this matrix to remove more than only one weight , because calculating the inverse Hess...

In this paper we have addressed the problem of unboundedness in the search direction when the Hessian is indefinite or near singular. A new algorithm has been proposed which naturally handles singular Hessian matrices, and is theoretically equivalent to the trust-region approach. This is accomplished by performing explicit matrix modifications adaptively that mimic the implicit modifications us...

Point process models such as the Epidemic-type Aftershock Sequence (ETAS) model have been widely used in the analysis and description of seismic catalogs and in shortterm earthquake forecasting. The standard errors of parameter estimates in the ETAS model are significant and cannot be ignored. This paper uses simulations to explore the accuracy of conventional standard error estimates based on ...