Classification and clustering are both important topics in statistical learning. A natural question herein is whether predefined classes are really different from one another, or whether clusters are really there. Specifically, we may be interested in knowing whether the two classes defined by some class labels (when they are provided), or the two clusters tagged by a clustering algorithm (where class labels are not provided), are from the same underlying distribution. Although both are challenging questions for the high-dimensional, low-sample size data, there has been some recent development for both. However, when it is costly to manually place labels on observations, it is often that only a small portion of the class labels is available. In this article, we propose a significance analysis approach for such type of data, namely partially labeled data. Our method makes use of the whole data and tries to test the class difference as if all the labels were observed. Compared to a testing method that ignores the label information, our method provides a greater power, meanwhile, maintaining the size, illustrated by a comprehensive simulation study. Theoretical properties of the proposed method are studied with emphasis on the high-dimensional, low-sample size setting. Our simulated examples help to understand when and how the information extracted from the labeled data can be effective. A real data example further illustrates the usefulness of the proposed method.

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.