Canopies: A Generalization of Vines and Vineyards for Parameterized Persistence

2026-06-22

AI Summary

This work addresses the challenge of characterizing multiplicity points and monodromy phenomena in parametrized persistent homology, which cannot be adequately captured by conventional persistence diagrams. The authors propose a novel “canopy” structure that lifts information from individual diagram points to the level of chain complexes by tracking the simplex pairs that generate them. This approach yields, for the first time, a combinatorial representation of multiplicity points and reveals their intrinsic connection to non-Hausdorff topological structures. Building upon filtered chain complexes, the framework introduces two variants—A-canopy and D-canopy—and leverages homeomorphism invariance to ensure stability. The method extends the vineyards formalism by enabling vine representations for multiplicity points and establishes a direct correspondence between monodromy and non-Hausdorff points within the canopy.

Abstract

In this paper, we provide a new construction for studying parameterized persistence, called a canopy. We give two versions of this construction: the A-canopy, retaining all information about points on the diagonal of the persistence diagram; and the D-canopy, encoding the information of the "standard" persistence diagram. We do this by making a simple but major modification in the persistence bundle representation information: namely, rather than tracking a point in the persistence diagram, we instead track some choice of pairs of simplices that created said point. This viewpoint is a combinatorial version of tracking the chain complex information rather than just the output of persistence. We show how to construct the canopies from any filtered filtration function, proving, using the algebraic structure of filtered chain complexes, that different choices of pairs result in homeomorphic structures. Finally, we showcase the power of our approach by using canopies to define vines even in the presence of points with multiplicity; to discuss monodromy; and to obtain some immediate results linking non-trivial monodromy in the persistent homology transform with the existence of non-Hausdorff points in the canopy.

Problem

• parameterized persistence
• persistence diagram
• monodromy
• multiplicity
• non-Hausdorff

Innovation

• canopy
• parameterized persistence
• persistent homology
• monodromy
• filtered chain complexes