  We compare two different linear dimensionality reduction strategies for the
multigroup classification problem: the trace ratio method and Fisher's
discriminant analysis. Recently, trace ratio optimization has gained in
popularity due to its computational efficiency, as well as the occasionally
better classification results. However, a statistical understanding is still
incomplete. We study and compare the properties of the two methods. Then, we
propose a robust version of the trace ratio method, to handle the presence of
outliers in the data. We reinterpret an asymptotic perturbation bound for the
solution to the trace ratio, in a contamination setting. Finally, we compare
the performance of the trace ratio method and Fisher's discriminant analysis on
both synthetic and real datasets, using classical and robust estimators.
