  This paper investigates the asymptotic behavior of the soft-margin and
hard-margin support vector machine (SVM) classifiers for simultaneously
high-dimensional and numerous data (large $n$ and large $p$ with
$n/p\to\delta$) drawn from a Gaussian mixture distribution. Sharp predictions
of the classification error rate of the hard-margin and soft-margin SVM are
provided, as well as asymptotic limits of as such important parameters as the
margin and the bias. As a further outcome, the analysis allow for the
identification of the maximum number of training samples that the hard-margin
SVM is able to separate. The precise nature of our results allow for an
accurate performance comparison of the hard-margin and soft-margin SVM as well
as a better understanding of the involved parameters (such as the number of
measurements and the margin parameter) on the classification performance. Our
analysis, confirmed by a set of numerical experiments, builds upon the convex
Gaussian min-max Theorem, and extends its scope to new problems never studied
before by this framework.
