A Nonlinear Dimensionality Reduction Framework Using Smooth Geodesics

Existing dimensionality reduction methods are adept at revealing hidden underlying manifolds arising from high-dimensional data and thereby producing a low-dimensional representation. However, the smoothness of the manifolds produced by classic techniques in the presence of noise is not guaranteed. In fact, the embedding generated using such non-smooth, noisy measurements may distort the geometry of the manifold and thereby produce an unfaithful embedding. Herein, we propose a framework for nonlinear dimensionality reduction that generates a manifold in terms of smooth geodesics that is designed to treat problems in which manifold measurements have been corrupted by noise. Our method generates a network structure for given high-dimensional data using a neighborhood search and then produces piecewise linear shortest paths that are defined as geodesics. Then, we fit points in each geodesic by a smoothing spline to emphasize the smoothness. The robustness of this approach for noisy and sparse datasets is demonstrated by the implementation of the method on synthetic and real-world datasets.