Reconstructing Embedded Graphs from Persistence Diagrams

Abstract: The persistence diagram (PD) is an increasingly popular topological descriptor. By encoding the size and prominence of topological features at varying scales, the PD provides important geometric and topological information about a space. Recent work has shown that well-chosen (finite) sets of PDs can differentiate between geometric simplicial complexes, providing a method for representing complex shapes using a finite set of descriptors. A related inverse problem is the following: given a set of PDs (or an oracle we can query for persistence diagrams), what is underlying geometric simplicial complex? In this paper, we present an algorithm for reconstructing embedded graphs in $\mathbb{R}^d$ (plane graphs in $\mathbb{R}^2$) with $n$ vertices from $n^2 - n + d + 1$ directional (augmented) PDs. Additionally, we empirically validate the correctness and time-complexity of our algorithm in $\mathbb{R}^2$ on randomly generated plane graphs using our implementation, and explain the numerical limitations of implementing our algorithm.