  In this paper, a novel framework of sparse kernel learning for Support Vector
Data Description (SVDD) based anomaly detection is presented. In this work,
optimal sparse feature selection for anomaly detection is first modeled as a
Mixed Integer Programming (MIP) problem. Due to the prohibitively high
computational complexity of the MIP, 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. The
QCLP-based iterative optimization problem is solved in a finite space called
the \emph{Empirical Kernel Feature Space} (EKFS) instead of in the input space
or \emph{Reproducing Kernel Hilbert Space} (RKHS). This is possible because of
the fact that 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.
Experimental results based on a hyperspectral image show that the proposed
method can provide improved performance over the current state-of-the-art
techniques.
