  We study the problem of finding the one-dimensional structure in a given data
set. In other words we consider ways to approximate a given measure (data) by
curves. We consider an objective functional whose minimizers are a
regularization of principal curves and introduce a new functional which allows
for multiple curves. We prove the existence of minimizers and establish their
basic properties. We develop an efficient algorithm for obtaining (near)
minimizers of the functional. While both of the functionals used are nonconvex,
we argue that enlarging the configuration space to allow for multiple curves
leads to a simpler energy landscape with fewer undesirable (high-energy) local
minima. Furthermore we note that the approach proposed is able to find the
one-dimensional structure even for data with considerable amount of noise.
