  Nonlinear dimensionality reduction (NLDR) algorithms such as Isomap, LLE and
Laplacian Eigenmaps address the problem of representing high-dimensional
nonlinear data in terms of low-dimensional coordinates which represent the
intrinsic structure of the data. This paradigm incorporates the assumption that
real-valued coordinates provide a rich enough class of functions to represent
the data faithfully and efficiently. On the other hand, there are simple
structures which challenge this assumption: the circle, for example, is
one-dimensional but its faithful representation requires two real coordinates.
In this work, we present a strategy for constructing circle-valued functions on
a statistical data set. We develop a machinery of persistent cohomology to
identify candidates for significant circle-structures in the data, and we use
harmonic smoothing and integration to obtain the circle-valued coordinate
functions themselves. We suggest that this enriched class of coordinate
functions permits a precise NLDR analysis of a broader range of realistic data
sets.
