Canopies: A Generalization of Vines and Vineyards for Parameterized Persistence
Provides a new theoretical tool for topological data analysis, enabling finer analysis of parameterized persistence, but is incremental as it extends existing bundle representations.
This paper introduces canopies, a new construction for parameterized persistence that tracks pairs of simplices creating persistence points rather than the points themselves. The authors show canopies enable defining vines with multiplicities and reveal links between monodromy and non-Hausdorff points.
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.