We study detection of collapse in high-dimensional point clouds, where mass concentrates near a lower-dimensional set relative to a non-collapsed geometry. We propose persistent homology-based test statistics under two well-studied filtrations, with cutoffs calibrated under a broad set of non-collapsed reference models. We benchmark power across three alternative collapse mechanisms (linear/spectral, nonlinear-support, and contamination/heterogeneity) and distill the results into a mechanism map guiding the choice of filtration and statistic.

翻译：我们研究高维点云中的坍缩检测问题，其中质量相对于非坍缩几何结构集中在低维集附近。我们基于两种经过充分研究的过滤方法提出持续同调检验统计量，其截断值在多种非坍缩参考模型下进行校准。我们针对三种替代坍缩机制（线性/谱坍缩、非线性支撑坍缩以及污染/异质性坍缩）进行功效比较，并将结果提炼为机制图谱以指导过滤方法和统计量的选择。