The geometry of nonparametric filament estimation

We consider the problem of estimating filamentary structure from d-dimensional point process data. We make some connections with computational geometry and develop nonparametric methods for estimating the filaments. We show that, under weak conditions, the filaments have a simple geometric representation as the medial axis of the data distribution's support. Our methods convert an estimator of the support's boundary into an estimator of the filaments. We also find the rates of convergence of our estimators. Proofs of all results are in the supplementary material available online. © 2012 American Statistical Association.