Rare class problems are common in real-world applications across a wide range of domains. Standard classification algorithms are known to perform poorly in these cases, since they focus on overall classification accuracy. In addition, we have seen a significant increase of data in recent years, resulting in many large scale rare class problems. In this paper, we focus on nonlinear kernel based classification methods expressed as a regularized loss minimization problem. We address the challenges associated with both rare class problems and large scale learning, by 1) optimizing area under curve of the receiver of operator characteristic in the training process, instead of classification accuracy and 2) using a rare class kernel representation to achieve an efficient time and space algorithm. We call the algorithm RankRC. We provide justifications for the rare class representation and experimentally illustrate the effectiveness of RankRC in test performance, computational complexity, and model robustness.

We consider the problem of hedging the loss of a given portfolio of derivatives using a set of more liquid derivative instruments. We illustrate why the typical mathematical formulation for this hedging problem is ill-posed. We propose to determine a hedging portfolio by minimizing a proportional cost subject to an upper bound on the hedge risk; this bound is typically slightly larger than the optimal hedge risk achievable without cost consideration. We illustrate that the optimal hedging portfolio obtained by the proposed method is attractive since it consists of fewer instruments with a comparable risk. Finally we illustrate the importance of modeling volatility uncertainty in hedge risk minimization.

ADMIT-1 enables the computation of sparse Jacobian and Hessian matrices, using automatic differentiation technology, from a MATLAB environment. Given a function to be differentiated, ADMIT-1 will exploit sparsity if present to yield sparse derivative matrices (in sparse MATLAB form). A generic automatic differentiation tool, subject to some functionality requirements, can be plugged into ADMIT-1; examples include ADOL-C (C/C++ target functions)and ADMAT (MATLAB target funcitons). ADMIT-1 also allows for the calculation of gradients and has several other related functions. This article provides an introduction to the design and usage of ADMIT-1.

In solving large sparse linear least squares problems A x ≃ b, several different numeric methods involve computing the same upper triangular factor R of A . It is of interest to be able to compute the nonzero structure of R , given only the structure of A . The solution to this problem comes from the theory of matchings in bipartite graphs. The structure of A is modeled with a bipartite graph, and it is shown how the rows and columns of A can be rearranged into a structure from which the structure of its upper triangular factor can be correctly computed. Also, a new method for solving sparse least squares problems, called block back-substitution, is presented. This method assures that no unnecessary space is allocated for fill, and that no unnecessary space is needed for intermediate fill.

The solution of a nonlinear optimization problem often requires an estimate of the Hessian matrix for a function f . In large scale problems, the Hessian matrix is usually sparse, and then estimation by differences of gradients is attractive because the number of differences can be small compared to the dimension of the problem. In this paper we describe a set of subroutines whose purpose is to estimate the Hessian matrix with the least possible number of gradient evaluations.

No abstract available.

In many nonlinear problems it is necessary to estimate the Jacobian matrix of a nonlinear mapping F. In large-scale problems the Jacobian of F is usually sparse, and then estimation by differences is attractive because the number of differences can be small compared with the dimension of the problem.For example, if the Jacobian matrix is banded, then the number of differences needed to estimate the Jacobian matrix is, at most, the width of the band.In this paper we describe a set of subroutines whose purpose is to estimate the Jacobian matrix of a mapping F with the least possible number of function evaluations.

No abstract available.