Recently, Dufrenois and Noyer proposed a one class Fisher's linear discriminant to isolate normal data from outliers. In this paper, a kernelized version of their criterion is presented. Originally on the basis of an iterative optimization process, alternating between subspace selection and clustering, I show here that their criterion has an upper bound making these two problems independent. In particular, the estimation of the label vector is formulated as an unconstrained binary linear problem (UBLP) which can be solved using an iterative perturbation method. Once the label vector is estimated, an optimal projection subspace is obtained by solving a generalized eigenvalue problem. Like many other kernel methods, the performance of the proposed approach depends on the choice of the kernel. Constructed with a Gaussian kernel, I show that the proposed contrast measure is an efficient indicator for selecting an optimal kernel width. This property simplifies the model selection problem which is typically solved by costly (generalized) cross-validation procedures. Initialization, convergence analysis, and computational complexity are also discussed. Lastly, the proposed algorithm is compared with recent novelty detectors on synthetic and real data sets.

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Linear discriminant analysis, such as Fisher's criterion, is a statistical learning tool traditionally devoted to separating a training dataset into two or even several classes by the way of linear decision boundaries. In this paper, we show that this tool can formalize the robust linear regression problem as a robust estimator will do. More precisely, we develop a one-class Fischer's criterion in which the maximization provides both the regression parameters and the separation of the data in two classes: typical data and atypical data or outliers. This new criterion is built on the statistical properties of the subspace decomposition of the hat matrix. From this angle, we improve the discriminative properties of the hat matrix which is traditionally used as outlier diagnostic measure in linear regression. Naturally, we call this new approach discriminative hat matrix. The proposed algorithm is fully nonsupervised and needs only the initialization of one parameter. Synthetic and real datasets are used to study the performance both in terms of regression and classification of the proposed approach. We also illustrate its potential application to image recognition and fundamental matrix estimation in computer vision.

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