# Reconstructing Embedded Graphs from Persistence Diagrams

arXiv: https://arxiv.org/abs/1912.08913 · HTML: https://mlsanity.com/paper/1912.08913/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 Dec 2019.

## Built on by (most cited first, 4 of 4)

- [Reconstruction of univariate functions from directional persistence diagrams](https://mlsanity.com/paper/2203.01894/) (2022)
- [Importance attribution in neural networks by means of persistence landscapes of time series](https://mlsanity.com/paper/2302.03132/) (2023)
- [Convolutional Persistence Transforms](https://mlsanity.com/paper/2208.02107/) (2022)
- [Exact Topological Compliance: Generating Persistence-Equivalent Graphs](https://mlsanity.com/paper/2610.18856/) (2026)

## Abstract (arXiv)

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 ℝd (plane graphs in ℝ2) with n vertices from n2 - n + d + 1 directional (augmented) PDs. Additionally, we empirically validate the correctness and time-complexity of our algorithm in ℝ2 on randomly generated plane graphs using our implementation, and explain the numerical limitations of implementing our algorithm.