Where Do Two Populations of Persistence Diagrams Differ? Calibrated Local Inference at a Fixed Budget
Abstract
Many two-sample tests for populations of persistence diagrams assess global differences without identifying the regions of the birth-death plane that contribute to them. We study simultaneous inference for local mean contrasts when the number of available diagrams is fixed. They are differences in expected weighted feature mass within ℓ∞ neighborhoods at several centers and radii. We estimate these contrasts using additive landmark responses. A Gaussian multiplier bootstrap calibrates simultaneous confidence intervals while allowing unequal group covariances. The neighborhoods whose intervals exclude zero form a map with approximate family-wise error control, and selecting a subset of original intervals for display preserves their joint coverage guarantee. On the simultaneous coverage event, every reported neighborhood lies within twice its radius of the support of the mean-measure difference. A geometric result gives sufficient radius conditions for a displaced feature to produce a nonzero contrast. A comparison of sufficient detection thresholds quantifies the tradeoff between reducing the number of tested coordinates and reserving observations for an independent pilot. In simulations with 40 to 120 diagrams per class, the bands achieved 94%-98% simultaneous coverage under both the strict null and equal means with unequal covariances. In the latter setting, a permutation maximum and the pooled-t implementation of the two-stage persistence-image test of Moon and Lazar rejected in up to 32% and 26% of runs, respectively. In the fixed-budget simulations, spending a third of the observations on a pilot to choose landmarks or radii located changes less often than a prespecified grid at a single radius. On the MUTAG benchmark, the localized region concentrates on rings of fused-ring systems, an exploratory reading.
Related
- Statistical Inference for Persistence Diagrams via Landmark Embeddings: Minimax Theory and Finite Approximation
- Persistence diagrams of finite topological spaces
- A result of continuous stability for generalized persistence diagrams
- On the Spectral Synthesis of Lipschitz Vectorizations of Persistence Diagrams
- Higher-order Persistence Diagrams
BibTeX
@misc{bagchi2026where,
title = {{Where Do Two Populations of Persistence Diagrams Differ? Calibrated Local Inference at a Fixed Budget}},
author = {Pramita Bagchi and Edward Bae and Atish Mitra and Alexander D. Silberman and Žiga Virk and Sushovan Majhi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.08292},
eprint = {2610.08292},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.08292},
}
RIS
TY - UNPB
TI - Where Do Two Populations of Persistence Diagrams Differ? Calibrated Local Inference at a Fixed Budget
AU - Pramita Bagchi
AU - Edward Bae
AU - Atish Mitra
AU - Alexander D. Silberman
AU - Žiga Virk
AU - Sushovan Majhi
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.08292
UR - https://arxiv.org/abs/2610.08292
AB - Many two-sample tests for populations of persistence diagrams assess global differences without identifying the regions of the birth-death plane that contribute to them. We study simultaneous inference for local mean contrasts when the number of available diagrams is fixed. They are differences in expected weighted feature mass within ℓ_∞ neighborhoods at several centers and radii. We estimate these contrasts using additive landmark responses. A Gaussian multiplier bootstrap calibrates simultaneous confidence intervals while allowing unequal group covariances. The neighborhoods whose intervals exclude zero form a map with approximate family-wise error control, and selecting a subset of original intervals for display preserves their joint coverage guarantee. On the simultaneous coverage event, every reported neighborhood lies within twice its radius of the support of the mean-measure difference. A geometric result gives sufficient radius conditions for a displaced feature to produce a nonzero contrast. A comparison of sufficient detection thresholds quantifies the tradeoff between reducing the number of tested coordinates and reserving observations for an independent pilot. In simulations with 40 to 120 diagrams per class, the bands achieved 94%-98% simultaneous coverage under both the strict null and equal means with unequal covariances. In the latter setting, a permutation maximum and the pooled-t implementation of the two-stage persistence-image test of Moon and Lazar rejected in up to 32% and 26% of runs, respectively. In the fixed-budget simulations, spending a third of the observations on a pilot to choose landmarks or radii located changes less often than a prespecified grid at a single radius. On the MUTAG benchmark, the localized region concentrates on rings of fused-ring systems, an exploratory reading.
ER -