The elements of the Hessian matrix consist of the second derivatives of the error measure with respect to the weights and thresholds in the network. They are needed in Bayesian estimation of network regularization parameters, for estimation of error bars on the network outputs, for network pruning algorithms, and for fast re-training of the network following a small change in the training data....

This paper presents an approach to estimate the uncertainty of registration parameters for the large displacement diffeomorphic metric mapping (LDDMM) registration framework. Assuming a local multivariate Gaussian distribution as an approximation for the registration energy at the optimal registration parameters, we propose a method to approximate the covariance matrix as the inverse of the Hes...

A family of probability distributions parametrized by an open domain Λ in Rn defines the Fisher information matrix on this domain which is positive semi-definite. In information geometry the standard assumption has been that the Fisher information matrix is positive definite defining in this way a Riemannian metric on Λ. If we replace the "positive definite" assumption by "0-deformable" conditi...

This paper extends factorized asymptotic Bayesian (FAB) inference for latent feature models (LFMs). FAB inference has not been applicable to models, including LFMs, without a specific condition on the Hessian matrix of a complete loglikelihood, which is required to derive a “factorized information criterion” (FIC). Our asymptotic analysis of the Hessian matrix of LFMs shows that FIC of LFMs has...

Let φ be a polynomial over K (a field of characteristic 0) such that the Hessian of φ is a nonzero constant. Let φ̄ be the formal Legendre Transform of φ. Then φ̄ is well-defined as a formal power series over K. The Hessian Conjecture introduced here claims that φ̄ is actually a polynomial. This conjecture is shown to be true when K = R and the Hessian matrix of φ is either positive or negative de...

We present a practical approach for solving volume and surface mesh optimization problems. Our approach is based on Newton’s method which uses both first-order (gradient) and second-order (Hessian) derivatives of the nonlinear objective function. The volume and surface optimization algorithms are modified such that surface constraints and mesh validity are satisfied. We also propose a simple an...

This study proposes a two-phase, algebraic approach to resolve the rotation cycle time and number of shipments for a multi-item, economic production quantity (EPQ) model with a random defective rate. A conventional method for solving a multi-item EPQ model is to use differential calculus and Hessian matrix equations on system cost function to prove convexity first and then derive an optimal pro...

Second-and higher-order derivatives are required by applications in scientiic computation, especially for optimization algorithms. The two complementary concepts of interpolating partial derivatives from univariate Taylor series and preaccumulating of \local" derivatives form the mathematical foundations for accurate, eecient computation of second-and higher-order partial derivatives for large ...

The question of the local stability of the ~replica-symmetric! amorphous solid state is addressed for a class of systems undergoing a continuous liquid to amorphous-solid phase transition driven by the effect of random constraints. The Hessian matrix, associated with infinitesimal fluctuations around the stationary point corresponding to the amorphous solid state, is obtained. The eigenvalues o...

This paper proposes some diagonal matrices that approximate the (inverse) Hessian by parts using the variational principle that is analogous to the one employed in constructing quasi-Newton updates. The way we derive our approximations is inspired by the least change secant updating approach, in which we let the diagonal approximation be the sum of two diagonal matrices where the first diagonal...