The ring of algebraic functions on persistence bar codes

Abstract

Persistent homology is a rapidly developing field in the study of numerous kinds of data sets. It is a functor which assigns to geometric objects so-called persistence bar codes, which are finite collections of intervals. These bar codes can be used to infer topological aspects of the geometric object. The set of all persistence bar codes, suitably defined, is known to possess metrics that are quite useful both theoretically and in practice. In this paper, we explore the possibility of coordinatizing, in a suitable sense, this same set of persistence bar codes. We derive a set of coordinates using results about multi-symmetric functions, study the property of the corresponding ring of functions, and demonstrate in an example how they work.