Unit canonical correlations and high-dimensional discriminant analysis

The problem of characterization of canonical vectors corresponding to unit sample canonical correlations in the non-full-rank case is considered. Classical work in this area is revisited and a new geometric characterization is developed. Applications are considered for classification problems that arise in the high-dimension low-sample-size setting. In that context, we show that Fisher's linear discriminant analysis and canonical correlation do not, in general, coincide and conduct empirical comparisons between the two methods.