Multi-manifold Discriminant Isomap for visualization and classification

Isomap aims to recover the intrinsic geometric structure of manifold by preserving geodesic distances between all pairs of data points. However it is an unsupervised dimensionality-reduction method. Usually, using class label information can increase the discriminating capability, hence a new supervised Isomap is proposed in this paper, dubbed Multi-manifold Discriminant Isomap (MMD-Isomap). First, data points are partitioned into different manifolds according to their class label information. Then, MMD-Isomap aims at seeking an optimal nonlinear subspace to preserve the geometrical structure of each manifold according to the Isomap criterion, meanwhile, to enhance the discriminating capability by maximizing the distances between data points of different manifolds. Finally, the corresponding optimization problem is solved by using a majorization algorithm. Furthermore, two new numerical metrics are designed to measure the performance of dimensionality-reduction method. In both visualization and classification experiments, MMD-Isomap achieves improved performance over many state-of-the-art methods.