  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 over sparse and noisy data is not guaranteed. In
fact, the embedding generated using such data 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 are either sparse or corrupted by noise. Our method generates a
network structure for given high-dimensional data using a nearest neighbors
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 sparse and noisy
datasets is demonstrated by the implementation of the method on synthetic and
real-world datasets.
