Calibrated Persistent Homology Tests for High-dimensional Collapse Detection
It offers a principled statistical framework for collapse detection in high-dimensional data, which is relevant for topological data analysis practitioners.
The paper develops calibrated persistent homology tests for detecting collapse in high-dimensional point clouds, and provides a mechanism map to guide filtration and statistic choice based on collapse type.
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.