Confidence Regions for Filamentary Structures

Filamentary structures, also called ridges, generalize the concept of modes of density functions and provide low-dimensional representations of point clouds. Using kernel type plug-in estimators, we give asymptotic confidence regions for filamentary structures based on two bootstrap approaches: multiplier bootstrap and empirical bootstrap. Our theoretical framework respects the topological structure of ridges by allowing the possible existence of intersections. Different asymptotic behaviors of the estimators are analyzed depending on how flat the ridges are, and our confidence regions are shown to be asymptotically valid in different scenarios in a unified form. As a critical step in the derivation, we approximate the suprema of the relevant empirical processes by those of Gaussian processes, which are degenerate in our problem and are handled by anti-concentration inequalities for Gaussian processes that do not require positive infimum variance.