Asymptotic and coarse-geometric behavior of COMPLEX

Characterize the asymptotic and coarse-geometric behavior of the COMPLEX slice-stack embedding on the unbounded space of multiparameter persistence modules.

Background

COMPLEX provides a closed-form embedding of multiparameter persistence modules by stacking certified landmark embeddings of one-parameter slices. The paper establishes finite-net distortion and prediction guarantees, while explicitly separating the large-scale behavior of the construction on the unbounded module space from the results proved here. Resolving this problem would determine whether the finite-dimensional certification framework has meaningful asymptotic or coarse-geometric properties beyond the bounded, finitely presented setting studied in the paper.

References

This paper develops it self-containedly on top of PLACE/PALACE~\citep{PaperI,PaperII}, whose population-level statistics---covariance estimation, two-sample tests and confidence sets for the mean embedding---are developed in a companion paper~\citep{PaperIII}; the asymptotic, coarse-geometric behavior of the same construction on the unbounded module space is a separate study we defer to future work.

— COMPLEX: A Closed-Form Certified Embedding of Multiparameter Persistence Modules  (2609.22012 - Majhi et al., 18 Sep 2026) in Section 1, Introduction