Semisupervised Multiclass Classification Problems With Scarcity of Labeled Data: A Theoretical Study

In recent years, the performance of semisupervised learning (SSL) has been theoretically investigated. However, most of this theoretical development has focused on binary classification problems. In this paper, we take it a step further by extending the work of Castelli and Cover to the multiclass paradigm. In particular, we consider the key problem in SSL of classifying an unseen instance x into one of K different classes, using a training data set sampled from a mixture density distribution and composed of l labeled records and u unlabeled examples. Even under the assumption of identifiability of the mixture and having infinite unlabeled examples, labeled records are needed to determine the K decision regions. Therefore, in this paper, we first investigate the minimum number of labeled examples needed to accomplish that task. Then, we propose an optimal multiclass learning algorithm, which is a generalization of the optimal procedure proposed in the literature for binary problems. Finally, we make use of this generalization to study the probability of error when the binary class constraint is relaxed.