Nonparametric Filament Estimation

We develop nonparametric methods for estimating filamentary structure from planar point process data and find the minimax lower bound for this problem. 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 find the rates of convergence of our estimators and show that when using an optimal boundary estimator, they achieve the minimax rate. Our work can be regarded as providing a solution to the manifold learning problem as well as being a new approach to principal curve estimation.