Persistent homology over the integers

Is it likely that in the future, there will be interest in computing persistent homology over the integers (or other PIDs)?

Currently, persistent homology is usually done over a field (like $\mathbb{Z}/2$), as the algorithms for producing the barcode only work for a field.

However, working over a field loses a lot of information. Also, there are algorithms that can compute the persistent groups over a PID (http://geometry.stanford.edu/papers/zc-cph-05/zc-cph-05.pdf Section 5).

Thanks for any help. References will be very welcome.

• Do you have any constraints on what you mean by an algorithm? In principle there's algorithms for all these types of computations but they're slow. Most of the "algorithm" papers have to do with either some sort of significant speed-up from the tautological constructions, or there's some other useful perspective, like stability, or parallelizability, or some new feature like having several gradings. For data from stats I doubt anyone would consider a double-exponential run-time "algorithm" to be a valid algorithm, as its too slow to be useful. – Ryan Budney Dec 19 '17 at 16:23
• Could you be a little more clear about what you're asking for? It's hard to give references since the question as stated could be answered with "Sure, there will probably be interest in the future, since there's a decent amount of interest in persistent homology now, but who can predict the future?" – j.c. Dec 19 '17 at 17:18
• This paper cites the Carlsson-Zomorodian paper and claims to show how to "infer the prime divisors of the torsion coefficients of the integral homology groups". – Jānis Lazovskis Dec 20 '17 at 7:42

1 Answer

As mentioned in Carlsson and Zomorodian's paper (to which you have linked), the problem of computing persistence barcodes with coefficients in a ring $R$ relies essentially on classifying graded modules over the polynomial ring $R[t]$. If (and only if!) $R$ is a field, $R[t]$ is a principal ideal domain and isomorphism classes of modules over it are easy to describe. When $R = \mathbb{Z}$, the modules are well-known to be hideous. Similar problems have plagued multi-dimensional persistence, since $R[s,t]$-modules are also pretty hard to classify (even when $R$ is a field).

But if you're willing to invest a bit in the Grothendieck construction, there might be hope: https://arxiv.org/abs/1601.03107