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 particular sets of PDs can differentiate between different shapes. This trait is desirable because it provides a method of representing complex shapes using finite sets of descriptors. The problem of choosing such a set of representative PDs and then using them to uniquely determine the shape is referred to as reconstruction. In this paper, we present an algorithm for reconstructing embedded graphs in R d (plane graphs in R 2 ) with n vertices from n 2 − n + d + 1 directional PDs. Lastly, we empirically validate the correctness and time-complexity of our algorithm in R 2 on randomly generated plane graphs using our implementation, and explain the numerical limitations of implementing our algorithm.