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：纤维结构（又称脊）推广了密度函数模态的概念，并为点云数据提供低维表示。基于核类型插入估计量，我们采用两种自助法（乘子自助法和经验自助法）构建纤维结构的渐近置信区域。我们的理论框架通过允许脊可能存在交叉点来尊重其拓扑结构，并根据脊的平坦程度分析了估计量的不同渐近行为，最终以统一形式证明了置信区域在不同场景下的渐近有效性。作为推导的关键步骤，我们将相关经验过程的上确界近似为高斯过程的上确界——这些高斯过程在本问题中呈现退化性，并通过无需假设正数下确界方差的高斯过程反集中不等式予以处理。