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.

翻译：摘要：丝状结构（也称为脊线）推广了密度函数模态的概念，并为点云提供了低维表示。基于核类型插入估计量，我们利用两种自助法（乘子自助法和经验自助法）给出了丝状结构的渐近置信区域。我们的理论框架通过允许交叉点的存在，尊重了脊线的拓扑结构。根据脊线平坦程度的不同，分析了估计量的不同渐近行为，并证明了我们的置信区域在不同场景下以统一形式渐近有效。在推导的关键步骤中，我们将相关经验过程的上确界近似为高斯过程的上确界——这些过程在我们的问题中是退化的，并通过无需正下确界方差的高斯过程反集中不等式进行处理。