Efficient cross-validation of kernel Fisher discriminant classifiers

Mika et al. [1] introduce a non-linear formulation of the Fisher discriminant based the well-known "kernel trick", later shown to be equivalent to the Least-Squares Support Vector Machine [2, 3]. In this paper, we show that the cross-validation error can be computed very efficiently for this class of kernel machine, specifically that leave-one-out cross-validation can be performed with a computational complexity of only O(l³) operations (the same as that of the basic training algorithm), rather than the O(l⁴) of a direct implementation. This makes leave-one-out cross-validation a practical proposition for model selection in much larger scale applications of KFD classifiers.