  Classification is an important tool with many useful applications. Among the
many classification methods, Fisher's Linear Discriminant Analysis (LDA) is a
traditional model-based approach which makes use of the covariance information.
However, in the high-dimensional, low-sample size setting, LDA cannot be
directly deployed because the sample covariance is not invertible. While there
are modern methods designed to deal with high-dimensional data, they may not
fully use the covariance information as LDA does. Hence in some situations, it
is still desirable to use a model-based method such as LDA for classification.
This article exploits the potential of LDA in more complicated data settings.
In many real applications, it is costly to manually place labels on
observations; hence it is often that only a small portion of labeled data is
available while a large number of observations are left without a label. It is
a great challenge to obtain good classification performance through the labeled
data alone, especially when the dimension is greater than the size of the
labeled data. In order to overcome this issue, we propose a semi-supervised
sparse LDA classifier to take advantage of the seemingly useless unlabeled
data. They provide additional information which helps to boost the
classification performance in some situations. A direct estimation method is
used to reconstruct LDA and achieve the sparsity; meanwhile we employ the
difference-convex algorithm to handle the non-convex loss function associated
with the unlabeled data. Theoretical properties of the proposed classifier are
studied. Our simulated examples help to understand when and how the information
extracted from the unlabeled data can be useful. A real data example further
illustrates the usefulness of the proposed method.
