Multi-Class Supervised Novelty Detection

In this paper we study the problem of finding a support of unknown high-dimensional distributions in the presence of labeling information, called Supervised Novelty Detection (SND). The One-Class Support Vector Machine (SVM) is a widely used kernel-based technique to address this problem. However with the latter approach it is difficult to model a mixture of distributions from which the support might be constituted. We address this issue by presenting a new class of SVM-like algorithms which help to approach multi-class classification and novelty detection from a new perspective. We introduce a new coupling term between classes which leverages the problem of finding a good decision boundary while preserving the compactness of a support with the l(2)-norm penalty. First we present our optimization objective in the primal and then derive a dual QP formulation of the problem. Next we propose a Least-Squares formulation which results in a linear system which drastically reduces computational costs. Finally we derive a Pegasos-based formulation which can effectively cope with large data sets that cannot be handled by many existing QP solvers. We complete our paper with experiments that validate the usefulness and practical importance of the proposed methods both in classification and novelty detection settings.