Nonlinear dimension reduction by local multidimensional scaling

Abstract

We propose a neighbourhood-preserving method called LMB for generating a low-dimensional representation of the data points scattered on a nonlinear manifold embedded in high-dimensional Euclidean space. Starting from an exemplary data point, LMB locally applies the classical Multidimensional Scaling (MDS) algorithm on small patches of the manifold and iteratively spreads the dimension reduction process. Differs to most dimension reduction methods, LMB does not require an input for the reduced dimension, as LMB could determine a well-fit dimension for reduction in terms of the pairwise distances of the data points. We thoroughly compare the performance of LMB with stateof-the-art linear and nonlinear dimension reduction algorithms on both synthetic data and real-world data. Numerical experiments show that LMB efficiently and effectively preserves the neighbourhood and uncovers the latent embedded structure of the manifold. LMB also has a low complexity of O(n2) for n data points.