Robin C. Gilbert

Robust support vector machines for classification and computational issues

In this paper, we investigate the theoretical and numerical aspects of robust classification using support vector machines (SVMs) by providing second order cone programming and linear programming formulations. SVMs are learning algorithms introduced by Vapnik used either for classification or regression. They show good generalization properties and they are based on statistical learning theory. The resulting learning problems are convex optimization problems suitable for application of primal-dual interior points methods. We investigate the training of a SVM in the case where a bounded perturbation is added to the value of an input x i ∈ℝ n . A robust SVM provides a decision function that is immune to data perturbations. We consider both cases where our training data are either linearly separable or non linearly separable respectively and provide computational results for real data sets.

Mathematical Foundations for Form Inspection and Adaptive Sampling

Nonlinear forms such as the cone, sphere, cylinder, and torus present significant problems in representation and verification. In this paper we examine linear and nonlinear forms using a heavily modified support vector machine (SVM) technique. The SVM approach applied to regression problems is used to derive quadratic programming problems that allow for generalized symbolic solutions to nonlinear regression. We have tested our approach to several geometries and achieved excellent results even with small data sets, making this method robust and efficient. More importantly, we identify process or inspection tendencies that could help in better designing the processes. Adaptive feature verification can be achieved through effective identification of the manufacturing pattern.

Form inspection using kernel methods

Abstract Form inspection of non-linear surfaces is a difficult task as suitable analytical models are often unavailable. This paper presents a mathematical model for surface inspection of face-milled plates and determination of the minimum zone based on a modification of the support vector machine (SVM) technique. The SVM approach is reformulated to regression problems using a different methodology than the ‘largest margin’ paradigm. In addition, this work derives extremely simple quadratic programming (QP) problems that allow for general symbolic solutions to non-linear regression problems. The results obtained from preliminary testing allow identification of processing tendencies so that a selective sampling procedure may be applied for inspecting future plates from that lot.

Quadratic programming formulations for classificationand regression

We reformulate the support vector machine approach to classification and regression problems using a different methodology than the classical ‘largest margin’ paradigm. From this, we are able to derive extremely simple quadratic programming problems that allow for general symbolic solutions to the classical problems of geometric classification and regression. We obtain a new class of learning machines that are also robust to the presence of small perturbations and/or corrupted or missing data in the training sets (provided that information about the amplitude of the perturbations is known approximately). A high performance framework for very large-scale classification and regression problems based on a Voronoi tessellation of the input space is also introduced in this work. Our approach has been tested on seven benchmark databases with noticeable gain in computational time in comparison with standard decomposition techniques such as SVM light .

Adaptive methods in coordinate metrology

The prudent selection of the sampling points ensures that representative points to typify a feature surface are obtained. The rationale is that the larger the number of sample points, the better the estimate of the surface. Large samples however lead to large measurement times and consequently time-induced errors. It is believed that a priori knowledge of process-induced errors can help in minimizing the total number of sampling points. Modeling the initial points for search, or approximate locations of errors is the key to minimizing the sampling effort. Suitable search methodology can then be used to determine the actual location of errors. If the regression surface describing the actually measured points can be identified, an adaptive search can be conducted. To do this we are using a kind of function learning machines that has been extensively developed the last decade, the Support Vector Regression (SVR). This paper describes in general terms our methodology.