Robust principal graphs for data approximation

Abstract

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.