  Manifold learning offers nonlinear dimensionality reduction of
high-dimensional datasets. In this paper, we bring geometry processing to bear
on manifold learning by introducing a new approach based on metric connection
for generating a quasi-isometric, low-dimensional mapping from a sparse and
irregular sampling of an arbitrary manifold embedded in a high-dimensional
space. Geodesic distances of discrete paths over the input pointset are
evaluated through "parallel transport unfolding" (PTU) to offer robustness to
poor sampling and arbitrary topology. Our new geometric procedure exhibits the
same strong resilience to noise as one of the staples of manifold learning, the
Isomap algorithm, as it also exploits all pairwise geodesic distances to
compute a low-dimensional embedding. While Isomap is limited to
geodesically-convex sampled domains, parallel transport unfolding does not
suffer from this crippling limitation, resulting in an improved robustness to
irregularity and voids in the sampling. Moreover, it involves only simple
linear algebra, significantly improves the accuracy of all pairwise geodesic
distance approximations, and has the same computational complexity as Isomap.
Finally, we show that our connection-based distance estimation can be used for
faster variants of Isomap such as L-Isomap.
