  While the existence of low-dimensional embedding manifolds has been shown in
patterns of collective motion, the current battery of nonlinear dimensionality
reduction methods are not amenable to the analysis of such manifolds. This is
mainly due to the necessary spectral decomposition step, which limits control
over the mapping from the original high-dimensional space to the embedding
space. Here, we propose an alternative approach that demands a two-dimensional
embedding which topologically summarizes the high-dimensional data. In this
sense, our approach is closely related to the construction of one-dimensional
principal curves that minimize orthogonal error to data points subject to
smoothness constraints. Specifically, we construct a two-dimensional principal
manifold directly in the high-dimensional space using cubic smoothing splines,
and define the embedding coordinates in terms of geodesic distances. Thus, the
mapping from the high-dimensional data to the manifold is defined in terms of
local coordinates. Through representative examples, we show that compared to
existing nonlinear dimensionality reduction methods, the principal manifold
retains the original structure even in noisy and sparse datasets. The principal
manifold finding algorithm is applied to configurations obtained from a
dynamical system of multiple agents simulating a complex maneuver called
predator mobbing, and the resulting two-dimensional embedding is compared with
that of a well-established nonlinear dimensionality reduction method.
