Sparse kernel learning-based feature selection for anomaly detection

In this paper, a novel framework of sparse kernel learning for support vector data description (SVDD) based anomaly detection is presented. By introducing 0-1 control variables to original features in the input space, sparse feature selection for anomaly detection is modeled as a mixed integer programming problem. Due to the prohibitively high computational complexity, it is relaxed into a quadratically constrained linear programming (QCLP) problem. The QCLP problem can then be practically solved by using an iterative optimization method, in which multiple subsets of features are iteratively found as opposed to a single subset. However, when a nonlinear kernel such as Gaussian radial basis function kernel, associated with an infinite-dimensional reproducing kernel Hilbert space (RKHS) is used in the QCLP-based iterative optimization, it is impractical to find optimal subsets of features due to a large number of possible combinations of the original features. To tackle this issue, a feature map called the empirical kernel map, which maps data points in the input space into a finite space called the empirical kernel feature space (EKFS), is used in the proposed work. The QCLP-based iterative optimization problem is solved in the EKFS instead of in the input space or the RKHS. This is possible because the geometrical properties of the EKFS and the corresponding RKHS remain the same. Now, an explicit nonlinear exploitation of the data in a finite EKFS is achievable, which results in optimal feature ranking. Comprehensive experimental results on three hyperspectral images and several machine learning datasets show that our proposed method can provide improved performance over the current state-of-the-art techniques.