Geometric Approach to Statistical Learning Theory through Support Vector Machines (SVM) with Application to Medical Diagnosis

This thesis deals with problems of Pattern Recognition in the framework of Machine Learning (ML) and, specifically, Statistical Learning Theory (SLT using Support Vector Machines (SVMs). The focus of this work is on the geometric interpretation of SVMs, which is accomplished through the notion of Reduced Convex Hulls (RCHs), and its impact on the derivation of new, efficient algorithms for the solution of the general SVM optimization task. The contributions of this work is the extension of the mathematical framework of RCHs, the derivation of novel geometric algorithms for SVMs and, finally, the application of the SVM algorithms to the field of Medical Image Analysis and Diagnosis (Mammography). Geometric SVM Framework’s extensions: The geometric interpretation of SVMs is based on the notion of Reduced Convex Hulls. Although the geometric approach to SVMs is very intuitive, its usefulness was restricted by the fact that the RCHs are defined as reduced convex combinations of the training points, thus, leading to combinatorial complexity of the corresponding optimization task. We extended the framework of RCHs with a set of theoretical results, that restrict the expression of the extreme points of the RCHs and provide an analytic form of their projection onto a specific direction. These results led to the derivation of novel, efficient geometric algorithms for SVMs. New SVM Algorithms: The known (and well-studied concerning convergence) geometric nearest point algorithms of, i) Gilbert and ii) Schlesinger-Kozinec, have been transformed (based on the above theoretical results) to solve the general, i.e., non-linear, non-separable SVM optimization task. These new geometric SVM algorithms have been implemented and tested against publicly available benchmark datasets and presented a clear performance advantage, compared to the fastest algebraic SVM algorithms. Applications – Mammography: The field of Medical Image Analysis and Diagnosis (and particularly of Mammography, studied in this work) is very crucial for social reasons and very demanding from the computational point of view. In this thesis, a set of qualitative and quantitative mammographic textural and morphological features have been assessed (using methods of statistical and fractal analysis); besides, several machine learning paradigms, e.g., Artificial Neural Networks (ANNs) and SVMs, have been used to discriminate benign from malignant mammographic masses. SVMs outperformed the other classifiers.