Uniqueness of models in persistent homology: the case of curves

TL;DR

This paper investigates whether persistent homology groups can uniquely identify generic curves in R^2 up to re-parameterization, providing conditions under which these topological invariants serve as unique characterizations.

Contribution

It establishes conditions under which persistent homology groups uniquely determine curves up to re-parameterization, and relates closeness of functions to the similarity of their persistent Betti numbers.

Findings

Persistent homology groups can characterize curves up to re-parameterization.

Equality of persistent homology groups implies the functions differ by a diffeomorphism.

Closeness of persistent Betti numbers corresponds to functions being close in max-norm.

Abstract

We consider generic curves in R^2, i.e. generic C^1 functions f from S^1 to R^2. We analyze these curves through the persistent homology groups of a filtration induced on S^1 by f. In particular, we consider the question whether these persistent homology groups uniquely characterize f, at least up to re-parameterizations of S^1. We give a partially positive answer to this question. More precisely, we prove that f=goh, where h:S^1-> S^1 is a C^1-diffeomorphism, if and only if the persistent homology groups of sof and sog coincide, for every s belonging to the group Sigma_2 generated by reflections in the coordinate axes. Moreover, for a smaller set of generic functions, we show that f and g are close to each other in the max-norm (up to re-parameterizations) if and only if, for every s belonging to Sigma_2, the persistent Betti numbers functions of sof and sog are close to each other, with respect to a suitable distance.