Persistent Sullivan Minimal Models of Metric Spaces

TL;DR

This paper introduces persistent Sullivan minimal models for topological data analysis, providing more discriminative invariants and sharper bounds for metric space comparisons than traditional persistent homology.

Contribution

It extends rational homotopy theory tools to persistent settings, establishing stability and new invariants for metric space analysis.

Findings

Interleaving distance between models is stable

New invariants outperform persistent homology in discrimination

Sharper Gromov-Hausdorff bounds using models

Abstract

We extend classical tools from rational homotopy theory to topological data analysis by introducing persistent Sullivan minimal models of persistent topological spaces. Our main result establishes that the interleaving distance between such models in the homotopy category of CDGAs is stable with respect to the homotopy interleaving distance of the underlying spaces. For Vietoris-Rips filtrations of metric spaces, this yields new persistent invariants that are more discriminative than persistent homology. We further show that these models provide sharper lower bounds for the Gromov-Hausdorff distance than those obtained from persistent homology or persistent rational homotopy groups.