The general theory required for the calculation of analytic third energy derivatives at the coupled-cluster level of theory is presented and connected to preceding special formulations for hyperpolarizabilities and polarizability gradients. Based on our theory, we have implemented a scheme for calculating the dipole Hessian matrix in a fully analytical manner within the coupled-cluster singles ...

In order to obtain more complete and continuous edge information of the image, the image enhancement methods and Hessian matrix are used in the process of edge detection. Based on the gradient information of color images, the pseudo-color edges can be got by using the multichannel edge detection. Then, enhance the edge information and remove the correlation to obtain complete edge information. ...

The dominant cost in solving least-square problems using Newton’s method is often that of factorizing the Hessian matrix over multiple values of the regularization parameter (λ). We propose an efficient way to interpolate the Cholesky factors of the Hessian matrix computed over a small set of λ values. This approximation enables us to optimally minimize the hold-out error while incurring only a...

In the neural-network parameter space, an attractive field is likely to be induced by singularities. In such a singularity region, first-order gradient learning typically causes a long plateau with very little change in the objective function value E (hence, a flat region). Therefore, it may be confused with “attractive” local minima. Our analysis shows that the Hessian matrix of E tends to be ...

Gaussian process (GP) regression is a Bayesian non-parametric regression model, showing good performance in various applications. However, it is quite rare to see research results on log-likelihood maximization algorithms. Instead of the commonly used conjugate gradient method, the Hessian matrix is first derived/simplified in this paper and the trust-region optimization method is then presente...

Second-order methods for neural network optimization have several advantages over methods based on first-order gradient descent, including better scaling to large mini-batch sizes and fewer updates needed for convergence. But they are rarely applied to deep learning in practice because of high computational cost and the need for model-dependent algorithmic variations. We introduce a variant of ...

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 ...

Recently Martens adapted the Hessian-free optimization method for the training of deep neural networks. One key aspect of this approach is that the Hessian is never computed explicitly, instead the Conjugate Gradient(CG) Algorithm is used to compute the new search direction by applying only matrix-vector products of the Hessian with arbitrary vectors. This can be done efficiently using a varian...

We consider sequential quadratic programming methods (SQP) globalized by linesearch for the standard exact penalty function. It is well known that if the Hessian of the Lagrangian is used in SQP subproblems, the obtained direction may not be of descent for the penalty function. The reason is that the Hessian need not be positive definite, even locally, under any natural assumptions. Thus, if a ...