Probabilistic linear discriminant analysis (PLDA) is a very effective feature extraction approach and has obtained extensive and successful applications in supervised learning tasks. It employs the squared$L_{2}$-norm to measure the model errors, which assumes a Gaussian noise distribution implicitly. However, the noise in real-life applications may not follow a Gaussian distribution. Particularly, the squared$L_{2}$-norm could extremely exaggerate data outliers. To address this issue, this article proposes a robust PLDA model under the assumption of a Laplacian noise distribution, called L1-PLDA. The learning process employs the approach by expressing the Laplacian density function as a superposition of an infinite number of Gaussian distributions via introducing a new latent variable and then adopts the variational expectation–maximization (EM) algorithm to learn parameters. The most significant advantage of the new model is that the introduced latent variable can be used to detect data outliers. The experiments on several public databases show the superiority of the proposed L1-PLDA model in terms of classification and outlier detection.

Locality preserving projection (LPP) is a well-known method for dimensionality reduction in which the neighborhood graph structure of data is preserved. Traditional LPP employ squared F-norm for distance measurement. This may exaggerate more distance errors, and result in a model being sensitive to outliers. In order to deal with this issue, we propose two novel F-norm-based models, termed as F-LPP and F-2DLPP, which are developed for vector-based and matrix-based data, respectively. In F-LPP and F-2DLPP, the distance of data projected to a low dimensional space is measured by F-norm. Thus it is anticipated that both methods can reduce the influence of outliers. To solve the F-norm-based models, we propose an iterative optimization algorithm, and give the convergence analysis of algorithm. The experimental results on three public databases have demonstrated the effectiveness of our proposed methods.