  This article considers the problem of sparse estimation of canonical vectors
in linear discriminant analysis when $p\gg N$. Several methods have been
proposed in the literature that estimate one canonical vector in the two-group
case. However, $G-1$ canonical vectors can be considered if the number of
groups is $G$. In the multi-group context, it is common to estimate canonical
vectors in a sequential fashion. Moreover, separate prior estimation of the
covariance structure is often required. We propose a novel methodology for
direct estimation of canonical vectors. In contrast to existing techniques, the
proposed method estimates all canonical vectors at once, performs variable
selection across all the vectors and comes with theoretical guarantees on the
variable selection and classification consistency. First, we highlight the fact
that in the $N>p$ setting the canonical vectors can be expressed in a closed
form up to an orthogonal transformation. Secondly, we propose an extension of
this form to the $p\gg N$ setting and achieve feature selection by using a
group penalty. The resulting optimization problem is convex and can be solved
using a block-coordinate descent algorithm. The practical performance of the
method is evaluated through simulation studies as well as real data
applications.
