Persistence Minimal Free Lie Model
Siheng Yi
TL;DR
This work merges rational homotopy theory with persistence modules to define persistence minimal Quillen models for rational $\mathbb{R}$-spaces. By applying Quillen's minimal free Lie model framework pointwise, it defines $M_{Qui}(\mathbb{X})$ so that each $M_{Qui}(\mathbb{X})_t$ is the minimal Quillen model of $\mathbb{X}_t$ and $M_{Qui}(\mathbb{X})_{s\le t}$ provides a Lie representative for $\mathbb{X}_{s\le t}$ up to weak equivalence, within $\mathbf{Ho}(\mathbf{DGL})$. The paper proves an instability bound chain $d_I^{\mathbf{Ho}(\mathbf{DGL})}(M_{Qui}(\mathbb{X}),M_{Qui}(\mathbb{Y})) \le d_{HI}(\mathbb{X},\mathbb{Y}) \le d_I(\mathbb{X},\mathbb{Y})$, with analogous relations for $H_*$ and $\pi_*$ persistence modules, establishing robust stability of the algebraic models. This algebraic perspective yields finer invariants than standard persistent homology and points toward future directions, such as persistence Lie-infinity models, for topological data analysis.
Abstract
The minimal Quillen model is a free Lie model for rational spaces proposed by Quillen. Meanwhile, persistence modules are theoretical abstractions of persistent homology. In this paper, we integrate the ideas of rational homotopy theory and persistence modules to construct the persistence minimal Quillen model and discuss its stability. Our results provide a new algebraic framework for topological data analysis, which is more refined compared to directly computing the homology groups of the filtration of simplicial complexes. Furthermore, the stability results for persistence minimal Lie models ensure that our model is well-founded.