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arXiv · 2110.14676

Algorithmic Reconstruction of the Fiber of Persistent Homology on Cell Complexes

Jacob Leygonie·Gregory Henselman-Petrusek
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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.

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Jacob Leygonie, Gregory Henselman-Petrusek. 2021-10-27. Algorithmic Reconstruction of the Fiber of Persistent Homology on Cell Complexes. https://arxiv.org/abs/2110.14676

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