Calibrated Persistent Homology Tests for High-dimensional Collapse Detection

TL;DR

This paper introduces calibrated persistent homology tests to detect high-dimensional collapse in point clouds, using two filtrations and benchmarking across multiple collapse mechanisms.

Contribution

It proposes a new calibration method for persistent homology-based tests and provides a mechanism map for choosing filtrations and statistics.

Findings

Calibrated tests effectively detect various collapse mechanisms.

Benchmarking shows the proposed methods outperform uncalibrated approaches.

Mechanism map guides filtration and statistic selection for collapse detection.

Abstract

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.