Supervised discriminant Isomap with maximum margin graph regularization for dimensionality reduction

Abstract

As one of the most popular nonlinear dimensionality reduction methods, Isomap has been widely used in pattern recognition and machine learning. However, Isomap has the following problems: (1) Isomap is an unsupervised dimensionality reduction method, it cannot use class label information to obtain discriminative low dimensional embedding for classification; (2) The embedding performance of Isomap is sensitive to neighborhood size parameter; (3) Isomap cannot deal with outside new data by direct embedding. In this paper, a novel dimensionality reduction method called supervised discriminant Isomap is proposed to solve the first two problems mentioned above. Specifically, first, raw data points are partitioned into different manifolds by using their class label information. Then, supervised discriminant Isomap aims at seeking an optimal nonlinear subspace to preserve the geometrical structure of each manifold according to the Isomap criterion, and to enhance the discriminating capability by maximizing the distances between data points of different classes and the maximum margin graph regularization term. Finally, the corresponding optimization problems are solved by using eigen-decomposition algorithm. Further, we extend supervised discriminant Isomap to a linear dimensionality reduction method called supervised discriminant Isomap projection for handling the above three problems. Moreover, our approaches have three important characteristics: (1) Proposed methods adaptively estimate the local neighborhood surrounding each sample based on data density and similarity; (2) The objective functions of proposed methods can maximize margins between the each classes in the dimension-reduced feature space; (3) The objective functions of proposed methods have closed-form solutions. Furthermore, our methods can capture more discriminative information from raw data than other Isomap based methods. Extensive experiments on nine data sets demonstrate that the proposed methods are superior to the related state-of-the-art methods.