Homology, Homotopy and Applications

Volume 18 (2016)

Number 1

The ring of algebraic functions on persistence bar codes

Pages: 381 – 402

DOI: https://dx.doi.org/10.4310/HHA.2016.v18.n1.a21


Aaron Adcock (Facebook, Inc., New York, N.Y., U.S.A.)

Erik Carlsson (Center of Mathematical Sciences and Applications, Harvard University, Cambridge, Massachusetts, U.S.A.)

Gunnar Carlsson (Department of Mathematics, Stanford University, Stanford, California, U.S.A.)


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.


persistent homology, point cloud, metric space, data analysis

2010 Mathematics Subject Classification

55N99, 62-07

Published 31 May 2016