  Research on manifold learning within a density ridge estimation framework has
shown great potential in recent work for both estimation and de-noising of
manifolds, building on the intuitive and well-defined notion of principal
curves and surfaces. However, the problem of unwrapping or unfolding manifolds
has received relatively little attention within the density ridge approach,
despite being an integral part of manifold learning in general. This paper
proposes two novel algorithms for unwrapping manifolds based on estimated
principal curves and surfaces for one- and multi-dimensional manifolds
respectively. The methods of unwrapping are founded in the realization that
both principal curves and principal surfaces will have inherent local maxima of
the probability density function. Following this observation, coordinate
systems that follow the shape of the manifold can be computed by following the
integral curves of the gradient flow of a kernel density estimate on the
manifold. Furthermore, since integral curves of the gradient flow of a kernel
density estimate is inherently local, we propose to stitch together local
coordinate systems using parallel transport along the manifold. We provide
numerical experiments on both real and synthetic data that illustrates clear
and intuitive unwrapping results comparable to state-of-the-art manifold
learning algorithms.
