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 PH takes a filter f:K rightarrow ℝ as input and returns the barcodes PH(f) of the associated sublevel set persistent homology modules. We address the inverse problem: given a target barcode D, computing the fiber PH-1(D). For this, we use the fact that 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 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.