L₂ Kernel Classification

Nonparametric kernel methods are widely used and proven to be successful in many statistical learning problems. Well--known examples include the kernel density estimate (KDE) for density estimation and the support vector machine (SVM) for classification. We propose a kernel classifier that optimizes the L₂ or integrated squared error (ISE) of a “difference of densities.” We focus on the Gaussian kernel, although the method applies to other kernels suitable for density estimation. Like a support vector machine (SVM), the classifier is sparse and results from solving a quadratic program. We provide statistical performance guarantees for the proposed L₂ kernel classifier in the form of a finite sample oracle inequality and strong consistency in the sense of both ISE and probability of error. A special case of our analysis applies to a previously introduced ISE-based method for kernel density estimation. For dimensionality greater than 15, the basic L₂ kernel classifier performs poorly in practice. Thus, we extend the method through the introduction of a natural regularization parameter, which allows it to remain competitive with the SVM in high dimensions. Simulation results for both synthetic and real-world data are presented.