  Various regularized linear discriminant analysis (LDA) methods have been
proposed to address the problems of the classic methods in high-dimensional
settings. Asymptotic optimality has been established for some of these methods
in high dimension when there are only two classes. A major difficulty in
proving asymptotic optimality for multiclass classification is that the
classification boundary is typically complicated and no explicit formula for
classification error generally exists when the number of classes is greater
than two. For the Fisher's LDA, one additional difficulty is that the
covariance matrix is also involved in the linear constraints. The main purpose
of this paper is to establish asymptotic consistency and asymptotic optimality
for our sparse Fisher's LDA with thresholded linear constraints in the
high-dimensional settings for arbitrary number of classes. To address the first
difficulty above, we provide asymptotic optimality and the corresponding
convergence rates in high-dimensional settings for a large family of linear
classification rules with arbitrary number of classes, and apply them to our
method. To overcome the second difficulty, we propose a thresholding approach
to avoid the estimate of the covariance matrix. We apply the method to the
classification problems for multivariate functional data through the wavelet
transformations.
