  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.
