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Algorithmic Reconstruction of the Fiber of Persistent Homology on Cell Complexes by Leygonie, Jacob; Henselman-Petrusek, Gregory is a scholarly article available to read on EtoBox.

What is Algorithmic Reconstruction of the Fiber of Persistent Homology on Cell Complexes about?

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.

Author

Leygonie, Jacob; Henselman-Petrusek, Gregory

Published

2021

Language

EN