A Difference Criterion for Dimensionality Reduction

A dynamic data-driven geoscience application includes hyperspectral scene classification which has shown promising potential in many remote-sensing applications. A hyperspectral image of a scene spectral radiance is typically measured by hundreds of contiguous spectral bands or features, ranging from visible/near-infrared (VNIR) to shortwave infrared (SWIR). Spectral-reflectance measurements provide rich information for object detection and classification. On the other hand, they generate a large number of features, resulting in a high dimensional measurement space. However, a large number of features often poses challenges and can result in poor classification performance. This is due to the curse of dimensionality which requires model reduction, uncertainty quantification and optimization for real-world applications. In such situations, feature extraction or selection methods play an important role by significantly reducing the number of features for building classifiers. In this work, we focus on efficient feature extraction using the dynamic data-driven applications systems (DDDAS) paradigm. Many dimension reduction techniques have been proposed in the literature. A well-known technique is Fisher's linear discriminant analysis (LDA). LDA finds the projection matrix that simultaneously maximizes a within class scatter matrix and minimizes a between class scatter matrix. However, LDA requires matrix inverse which can be a major issue when the within matrix is singular. We propose a difference criterion for dimension reduction that does not require a matrix inverse for software implementation. We show how to solve the optimization problem with semi-definite programming. In addition, we establish an error bound for the proposed algorithm. We demonstrate the connection between relief feature selection and a two class formulation of multi-class problems, thereby providing a sound basis for observed benefits associated with this formulation. Finally, we provide experimental results demonstrating that the proposed technique is competitive against competing methods in a number of hyperspectral image data that is broadly applicable to geoscience applications.