Geometry and statistics-preserving manifold embedding for nonlinear dimensionality reduction

Nonlinear dimensionality reduction based on preservation of geometry is an imperative task in many scientific applications such as face recognition, object detection, data visualization, and classification. The traditional methods such as Isomap, locally linear embedding (LLE), and Laplacian eigen map (LEM) preserve local or global geometry or both with several limitations such as topological instability in the presence of noise or large neighborhood, inability to deal with ill-connected data points, high sensitivity of embedding performance to selected neighbor number. They also fail to retain the statistical properties of the data such as mean and variance while reducing the data dimensionality. In this paper, we propose a novel method, referred to as geometry and statistics-preserving embedding (GSE), which preserves both geometry and statistics of the data as well as overcomes some critical limitations of the above-mentioned methods. Moreover, GSE offers nonlinear mapping based on position information of the geometry, which is not possible by any other available method. We run experiments with GSE on several datasets including facial images and single cell RNA sequence data, where the GSE has been found to provide significantly better manifold embedding results than the traditional methods. The implementation codes of GSE are available at https://github.com/tauhidstanford/GSE.