Limit Theorems for Verbose Persistence Diagrams

Abstract

The persistence diagram is a central object in the study of persistent homology and has also been investigated in the context of random topology. The more recent notion of the verbose diagram (a.k.a. verbose barcode) is a refinement of the persistence diagram. Whereas the persistence diagram is a complete invariant of persistent homology, the verbose persistence diagram is a complete invariant of the one-level higher object -- a filtered chain complex. It therefore strictly contains the persistence diagram, both in form and in the amount of information it encodes. Concretely, the verbose diagram incorporates ephemeral features that arise in a filtered topological space, representing them as additional points along the diagonal. In this work, we initiate the study of random verbose diagrams. We establish a strong law of large numbers for verbose diagrams as a random point cloud grows in size -- that is, we prove the existence of a limiting verbose diagram, viewed as a measure on the half-plane on and above the diagonal. Also, we characterize its support and compute its total mass. Along the way, we extend the notion of the persistent Betti number, reveal the relation between this extended notion and the verbose diagram (which is an extension of the fundamental lemma of persistent homology), and establish results on the asymptotic behavior of the extended persistent Betti numbers. This work extends the main results of the work by Hiraoka, Shirai, and Trinh and its sequel by Shirai and Suzaki to the setting of verbose diagrams.