  Revealing hidden geometry and topology in noisy data sets is a challenging
task. Elastic principal graph is a computationally efficient and flexible data
approximator based on embedding a graph into the data space and minimizing the
energy functional penalizing the deviation of graph nodes both from data points
and from pluri-harmonic configuration (generalization of linearity). The
structure of principal graph is learned from data by application of a
topological grammar which in the simplest case leads to the construction of
principal curves or trees. In order to more efficiently cope with noise and
outliers, here we suggest using a trimmed data approximation term to increase
the robustness of the method. The modification of the method that we suggest
does not affect either computational efficiency or general convergence
properties of the original elastic graph method. The trimmed elastic energy
functional remains a Lyapunov function for the optimization algorithm. On
several examples of complex data distributions we demonstrate how the robust
principal graphs learn the global data structure and show the advantage of
using the trimmed data approximation term for the construction of principal
graphs and other popular data approximators.
