Algorithmic reconstruction of the fiber of persistent homology on cell complexes
Jacob Leygonie, Gregory Henselman‐Petrusek · Journal of Applied and Computational Topology · 2024
Abstract Let K be a finite simplicial, cubical, delta or CW complex. The persistence map $$\textrm{PH}$$ PH takes a filter $$f:K\rightarrow \mathbb {R}$$ f : K → 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 $$\textrm{PH}^{-1}(D)$$ PH - 1 ( D ) . For this, we use the fact that $$\textrm{PH}^{-1}(D)$$ 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 $$\textrm{PH}^{-1}(D)$$ 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.