Algorithmic Reconstruction of the Fiber of Persistent Homology on Cell Complexes

TL;DR

This paper develops an algorithmic method to reconstruct the set of all filters on cell complexes that produce a given persistent homology barcode, extending previous results to general CW complexes and analyzing their statistical properties.

Contribution

It generalizes the fiber reconstruction of persistent homology from simplicial complexes to arbitrary CW complexes and introduces a depth-first search algorithm for this purpose.

Findings

Successfully reconstructed fibers for 120 sample problems.

Provided initial statistical analysis of fiber structures.

Extended theoretical framework to general CW complexes.

Abstract

Let $K$ be a finite simplicial, cubical, delta or CW complex. The persistence map $\mathrm{PH}$ takes a filter $f:K \rightarrow \mathbb{R}$ as input and returns the barcodes $\mathrm{PH}(f)$ of the associated sublevel set persistent homology modules. We address the inverse problem: given a target barcode $D$, computing the fiber $\mathrm{PH}^{-1}(D)$. For this, we use the fact that $\mathrm{PH}^{-1}(D)$ decomposes as complex of polyhedra when $K$ is a simplicial complex, and we generalise this result to arbitrary based chain complexes. We then design and implement a depth first search algorithm that recovers the polyhedra forming the fiber $\mathrm{PH}^{-1}(D)$. As an application, we solve a corpus of 120 sample problems, providing a first insight into the statistical structure of these fibers, for general CW complexes.