Kernel Ellipsoidal Trimming

Ellipsoid estimation is an issue of primary importance in many practical areas such as control, system identification, visual/audio tracking, experimental design, data mining, robust statistics and novelty/outlier detection. This paper presents a new method of kernel information matrix ellipsoid estimation (KIMEE) that finds an ellipsoid in a kernel defined feature space based on a centered information matrix. Although the method is very general and can be applied to many of the aforementioned problems, the main focus in this paper is the problem of novelty or outlier detection associated with fault detection. A simple iterative algorithm based on Titterington's minimum volume ellipsoid method is proposed for practical implementation. The KIMEE method demonstrates very good performance on a set of real-life and simulated datasets compared with support vector machine methods

Statistical properties of the method of regularization with periodic Gaussian reproducing kernel

The method of regularization with the Gaussian reproducing kernel is popular in the machine learning literature and successful in many practical applications. In this paper we consider the periodic version of the Gaussian kernel regularization. We show in the white noise model setting, that in function spaces of very smooth functions, such as the infinite-order Sobolev space and the space of analytic functions, the method under consideration is asymptotically minimax; in finite-order Sobolev spaces, the method is rate optimal, and the efficiency in terms of constant when compared with the minimax estimator is reasonably high. The smoothing parameters in the periodic Gaussian regularization can be chosen adaptively without loss of asymptotic efficiency. The results derived in this paper give a partial explanation of the success of the Gaussian reproducing kernel in practice. Simulations are carried out to study the finite sample properties of the periodic Gaussian regularization.Comment: Published by the Institute of Mathematical Statistics (http://www.imstat.org) in the Annals of Statistics (http://www.imstat.org/aos/) at http://dx.doi.org/10.1214/00905360400000045

Comprehensible credit scoring models using rule extraction from support vector machines.

In recent years, Support Vector Machines (SVMs) were successfully applied to a wide range of applications. Their good performance is achieved by an implicit non-linear transformation of the original problem to a high-dimensional (possibly infinite) feature space in which a linear decision hyperplane is constructed that yields a nonlinear classifier in the input space. However, since the classifier is described as a complex mathematical function, it is rather incomprehensible for humans. This opacity property prevents them from being used in many real- life applications where both accuracy and comprehensibility are required, such as medical diagnosis and credit risk evaluation. To overcome this limitation, rules can be extracted from the trained SVM that are interpretable by humans and keep as much of the accuracy of the SVM as possible. In this paper, we will provide an overview of the recently proposed rule extraction techniques for SVMs and introduce two others taken from the artificial neural networks domain, being Trepan and G-REX. The described techniques are compared using publicly avail- able datasets, such as Ripley's synthetic dataset and the multi-class iris dataset. We will also look at medical diagnosis and credit scoring where comprehensibility is a key requirement and even a regulatory recommendation. Our experiments show that the SVM rule extraction techniques lose only a small percentage in performance compared to SVMs and therefore rank at the top of comprehensible classification techniques.Credit; Credit scoring; Models; Model; Applications; Performance; Space; Decision; Yield; Real life; Risk; Evaluation; Rules; Neural networks; Networks; Classification; Research;

Two qubit copying machine for economical quantum eavesdropping

We study the mapping which occurs when a single qubit in an arbitrary state interacts with another qubit in a given, fixed state resulting in some unitary transformation on the two qubit system which, in effect, makes two copies of the first qubit. The general problem of the quality of the resulting copies is discussed using a special representation, a generalization of the usual Schmidt decomposition, of an arbitrary two-dimensional subspace of a tensor product of two 2-dimensional Hilbert spaces. We exhibit quantum circuits which can reproduce the results of any two qubit copying machine of this type. A simple stochastic generalization (using a ``classical'' random signal) of the copying machine is also considered. These copying machines provide simple embodiments of previously proposed optimal eavesdropping schemes for the BB84 and B92 quantum cryptography protocols.Comment: Minor changes. 26 pages RevTex including 7 PS figure