  This article carries out a large dimensional analysis of standard regularized
discriminant analysis classifiers designed on the assumption that data arise
from a Gaussian mixture model with different means and covariances. The
analysis relies on fundamental results from random matrix theory (RMT) when
both the number of features and the cardinality of the training data within
each class grow large at the same pace. Under mild assumptions, we show that
the asymptotic classification error approaches a deterministic quantity that
depends only on the means and covariances associated with each class as well as
the problem dimensions. Such a result permits a better understanding of the
performance of regularized discriminant analsysis, in practical large but
finite dimensions, and can be used to determine and pre-estimate the optimal
regularization parameter that minimizes the misclassification error
probability. Despite being theoretically valid only for Gaussian data, our
findings are shown to yield a high accuracy in predicting the performances
achieved with real data sets drawn from the popular USPS data base, thereby
making an interesting connection between theory and practice.
