Algorithmic reconstruction of the fiber of persistent homology on cell complexes

Let K be a finite simplicial, cubical, delta or CW complex. The persistence map  \(\textrm{PH}\) $\text{PH}$ takes a filter  \(f:K\rightarrow \mathbb {R}\) $f:K\to \mathbb{R}$ as input and returns the barcodes of the sublevel set persistent homology of f in each dimension. We address the inverse problem: given target barcodes D, computing the fiber  \(\textrm{PH}^{-1}(D)\) ${\text{PH}}^{-1}(D)$ . For this, we use the fact that  \(\textrm{PH}^{-1}(D)\) ${\text{PH}}^{-1}(D)$ decomposes as a polyhedral complex 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 that recovers the polytopes forming the fiber  \(\textrm{PH}^{-1}(D)\) ${\text{PH}}^{-1}(D)$ . 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.