Density-based multi-manifold ISOMAP for data classification

ISOMAP is a widely used unsupervised nonlinear dimensionality reduction algorithm, which preserves the coordinate transformation of high-dimensional space to low- dimensional space by keeping the geodesic distance between the observation samples. It can find the intrinsic low-dimensional structure hidden in high-dimensional data uniformly distributed on or around a single manifold. However, it is an unsupervised dimensionality-reduction method. Usually, for the classification task, using class label can increase the discriminating capability. Moreover, if the data are sampled from multi-class, each of which corresponds to an independent manifold, several disconnected neighborhood graphs may form, which leads to the failure of ISOMAP algorithm. In this paper, a novel supervised ISOMAP based on density is proposed, dubbed Density-based Multi-Manifold ISOPMAP (DMM-ISOMAP). Firstly, we choose appropriate value, and construct a single neighborhood graph for each class by using class label, with which short-circuit edges cannot be produced. Then find the density peak point for each graph and connect them. Finally, the interclass edges are multiplied by a scaling factor that is larger than one to enhance the discrimination ability by magnifying the distances between data points of different manifold. Meanwhile, a bigger single neighborhood graph is incorporated and then ISOMAP algorithm is applied to find the intrinsic low-dimensional embedding structure. The experimental results for classification tasks on synthetic and real-world data confirm the effectiveness of the proposed method.