Component preserving laplacian eigenmaps for data reconstruction and dimensionality reduction

Laplacian Eigenmaps (LE) is a widely used dimensionality reduction and data reconstruction method. When the data has multiple connected components, the LE method has two obvious deficiencies. First, it might reconstruct each component as a single point, resulting in loss of information within the component. Second, it only focuses on local features but ignores the location information between components, which might cause the reconstructed components to overlap or to completely change their relative positions. To solve these two problems, this article first modifies the optimization objective of the LE method, by characterizing the relative positions between components of data with the similarity between high-density core points, and then solves the optimization problem by using a gradient descent method to avoid the over-compression of data points in the same connected component. A series of experiments on synthetic data and real-world data verify the effectiveness of the proposed method.