Unsupervised Feature Selection Based on Ultrametricity and Sparse Training Data: A Case Study for the Classification of High-Dimensional Hyperspectral Data

In this paper, we investigate the potential of unsupervised feature selection techniques for classification tasks, where only sparse training data are available. This is motivated by the fact that unsupervised feature selection techniques combine the advantages of standard dimensionality reduction techniques (which only rely on the given feature vectors and not on the corresponding labels) and supervised feature selection techniques (which retain a subset of the original set of features). Thus, feature selection becomes independent of the given classification task and, consequently, a subset of generally versatile features is retained. We present different techniques relying on the topology of the given sparse training data. Thereby, the topology is described with an ultrametricity index. For the latter, we take into account the Murtagh Ultrametricity Index (MUI) which is defined on the basis of triangles within the given data and the Topological Ultrametricity Index (TUI) which is defined on the basis of a specific graph structure. In a case study addressing the classification of high-dimensional hyperspectral data based on sparse training data, we demonstrate the performance of the proposed unsupervised feature selection techniques in comparison to standard dimensionality reduction and supervised feature selection techniques on four commonly used benchmark datasets. The achieved classification results reveal that involving supervised feature selection techniques leads to similar classification results as involving unsupervised feature selection techniques, while the latter perform feature selection independently from the given classification task and thus deliver generally versatile features