  The classification of high dimensional data with kernel methods is considered
in this article. Exploit- ing the emptiness property of high dimensional
spaces, a kernel based on the Mahalanobis distance is proposed. The computation
of the Mahalanobis distance requires the inversion of a covariance matrix. In
high dimensional spaces, the estimated covariance matrix is ill-conditioned and
its inversion is unstable or impossible. Using a parsimonious statistical
model, namely the High Dimensional Discriminant Analysis model, the specific
signal and noise subspaces are estimated for each considered class making the
inverse of the class specific covariance matrix explicit and stable, leading to
the definition of a parsimonious Mahalanobis kernel. A SVM based framework is
used for selecting the hyperparameters of the parsimonious Mahalanobis kernel
by optimizing the so-called radius-margin bound. Experimental results on three
high dimensional data sets show that the proposed kernel is suitable for
classifying high dimensional data, providing better classification accuracies
than the conventional Gaussian kernel.
