  We present in this paper an application of the theory of principal bundles to
the problem of nonlinear dimensionality reduction in data analysis. More
explicitly, we derive, from a 1-dimensional persistent cohomology computation,
explicit formulas for circle-valued functions on data with nontrivial
underlying topology. We show that the language of principal bundles leads to
coordinates defined on an open neighborhood of the data, but computed using
only a smaller subset of landmarks. It is in this sense that the coordinates
are sparse. Several data examples are presented, as well as theoretical results
underlying the construction.
